UNIVERSITY  OF  CALIFORNIA. 

GIFT  OK  * 

(#^4    #.^-*fe._.lHft 


Accessions 


Shelf  No. 


/;   '       .'.-••  ' 

' 


• 


/* 


^^c-x     s^.c£j_ 

^~£^*—G-jt^ 

/U--«L-tf 
Onrix^ts   >-#>*^ 


X^-C^LC^ 


/  / 


"1 


v  P    Usctn'ts 


THIS  volume,  announced  several  years  ago  as  being  in 
preparation  to  follow  the  author's  Land  Surveying,  was  left 
at  the  time  of  his  death  unfinished.  In  preparing  it  for  the 
press,  the  editor  has  endeavored  to  carry  out,  as  far  as  pos- 
sible, the  plan  already  laid  down.  A  considerable  part  of  the 
volume  has  been  given  by  the  author,  in  the  form  of  lectures 
to  the  civil  engineering  classes  in  Union  College,  and  has 
been  printed  from  the  original  manuscript. 

The  best  authors  on  the  subjects  treated  have  been  con- 
sulted, in  order  to  render  the  work  as  complete  as  possible. 
The  principal  authorities  are  the  following:  Begat,  Bourns, 
Breton,  Chauvenet,  Fenwick,  Frome,  Gurley,  Guyot,  Jackson 
Lee,  Narrien,  Puissant,  Salneuve,  Simms,  Smith,  Stephenson, 
Wiesbach,  Williams,  and  the  papers  of  the  United  States 
Coast  Survey. 

Wherever  it  has  been  necessary  to  refer  to  the  elementary 
principles  of  surveying,  reference  has  been  made  to  Gillespie's 
Land  Surveying  (designated  as  L.  S.  for  brevity),  and  the 
numbers  of  the  articles  referred  to  are  enclosed  in  parentheses. 

UNION  COLLEGE, 
SCHENECTADY,  June,  1870. 


ANALYTICAL  TABLE  OF  CONTENTS. 


INTRODUCTION. 

ART.  PAGE 

1.  Levelling  in  General 1 

2.  Direct  Levelling 1 

8.  Indirect  Levelling. . 2 

4.  Barometric  Levelling 2 

6.  Topography 2 

6.  Special  Objects  and  Difficulties 3 

7.  Underground  Surveying 3 

8.  "Water  Surveying 3 

9.  Reflecting  Instruments 3 

10.  Spherical  Surveying 3 

11.  Location. .  .4 


PART  I. 

DIRECT    LEVELLING. 

CHAPTER    I. — GENERAL    PRINCIPLES. 

12.  Levelling  Instruments  6 

13.  Methods  of  Operation 5 

14.  Curvature 6 

15.  Refraction , 7 

CHAPTER    H. — PERPENDICULAR    LEVELS. 

16.  Principle 7 

17.  Plumb-line  Levels 7 

18.  Reflecting  Levels 8 


ANALYTICAL  TABLE  OF  CONTENTS. 


CHAPTER    HI. — WATER-LEVELS. 

AET.  PAGE 

19.  Continuous  Water-levels 10 

20.  Visual  Water-levels..  .   11 


CHAPTER   IV. — AIR   BUBBLE   OR   SPIRIT   LEVELS. 

21.  The  Spirit-level 12 

22.  Sensibility 12 

23.  Block-level 13 

24.  Level  with  Sights 14 

25.  Hand  Reflected  Leyel 14 

26.  The  Telescope  Level 15 

27.  The  Y  Level 16 

28.  The  Telescope. . . 16 

29.  The  Cross-hairs . . . . 17 

30.  The  Level ...  17 

31.  Supports 18 

32.  Parallel  Plates 18 

33.  Adjustments 18 

34.  First  Adjustment .  19 

35.  Second  Adjustment ^ 20 

36.  Third  Adjustment 20 

37.  Centring  the  Object-glass  and  Eye-piece 21 

38.  Adjustment  by  the  "  Peg  Method  " 22 

39.  Verification  by  another  Telescope 23 

40.  Egault's  Level 23 

41.  Troughton's  Level ' 23 

42.  Gravatt's  Level 24 

43.  Bonrdal6ue's  Level 24 

44.  Lenoir's  Level 24 

45.  Tripods 25 


CHAPTER  V. — RODS. 

46.  How  made 25 

47.  Target 26 

48.  Vernier's 27 

49.  New-York  Kod 27 

50.  Boston  Rod 27 

51.  Speaking  Rods , 28 


ANALYTICAL  TABLE  OF  CONTENTS. 


CHAPTER. VI. THE   PRACTICE. 

AKT.  PAGE 

52.  Field  Routine 30 

53.  Field-notes 32 

54.  First  Form  of  Field-book 33 

55.  Second  Form  of  Field-book. ...  35 

56.  Third  Form  of  Field-book 38 

67.  Best  Length  of  Sight ' '. 39 

58.  Equal  Distances  of  Sight 39 

59.  Datum-level 39 

60.  Bench-marks 40 

61.  Test-levels 40 

62.  Limits  of  Precision 41 

63.  Flying-levels * 41 

64.  Levelling  for  Sections ' 41 

65.  Profiles 41 

66.  Cross-levels \ ..  .....  .42 


CHAPTER   VII. — DIFFICULTIES. 

67.  Steep  Slopes 43 

68.  When  the  Eod  is  too  low 44 

69.  When  the  Rod  is  too  high 45 

70.  When  the  Rod  is  too  near 45 

71.  Levelling  across  Water 45 

72.  Across  a  Swamp  or  Marsh 46 

73.  Through  Underwood 46 

74.  Over  a  Board  Fence 46 

75.  Over  a  Wall. 46 

76.  Through  a  House 47 

77.  The  Sun 47 

78.  Wind 48 

79.  Idiosyncrasies 48 

80.  Reciprocal  Levelling 48 


CHAPTER   VIH. — LEVELLING   LOCATION. 

81.  Its  Nature 49 

82.  Difficulties 49 

83.  Staking  out  Work , 50 

84.  To  Locate  a  Level  Line 51 

85.  Applications 51 

86.  To  run  a  Grade  Line  . .  .52 


Viii  ANALYTICAL  TABLE  OF  CONTENTS, 

PART    II. 
INDIRECT   LEVELLING. 

CHAPTER    I. METHODS   AND   INSTRUMENTS. 

ART.  PAGE. 

87.  Vertical  Surveying 53 

88.  Vertical  Angles .• 54 

89.  Instruments 55 

90.  Slopes 56 

91.  Theodolites 56 

92.  Surveyor's  Transit 66 

93.  Adjustments '58 

94.  Field- Work 60 

95.  Angular  Profiles 61 

96.  Burnier's  Level 62 

97.  German  Universal  Instrument 62 

CHAPTER   II. — SIMPLE   ANGULAR   LEVELLING. 

A. — For  /Short  Distances. 

98.  Principle 63 

99.  Best-conditioned  Angle 63 

B. — For  Greater  Distances. 

100.  Correction  for  Curvature 64 

101.  Correcting  the  Eesult 64 

102.  Correcting  the  Angle 64 

103.  Correction  for  Refraction 64 

C. — For  Very  Great  Distances. 

104.  Correction  fbr  Curvature 65 

105.  Correction  for  Refraction 66 

106.  Reciprocal  Observation  for  cancelling  Refraction 67 

107.  Reduction  to  the  Summits  of  the  Signals t 67 

108.  When  the  Height  of  the  Signal  cannot  be  Measured 68 

109.  Levelling  by  the  Horizon  of  the  Sea 69 

CHAPTER   HI. COMPOUND   ANGULAR   LEVELLING. 

110.  By  Angular  Coordinates  in  one  Plane. 70 

111.  By  Angular  Coordinates  in  several  Planes 71 

112.  Conversely 71 


ANALYTICAL  TABLE  OF  CONTEXTS.  ix 

PAKT    III. 

BAROMETRIC   LEVELLING. 

CHAPTER   I. — PRINCIPLES   AND  FORMULAS. 

AST.  PAGE 

113.  Principles 73 

114.  Applications 73 

115.  Correction  for  Temperature  of  the  Mercury 74 

116.  Correction  for  Temperature  of  the  Air 74 

117.  Other  Corrections. 74 

118.  Eules  for  calculating  Heights 75 

119.  Formulas 75 

120.  To  Correct  for  Latitude 76 

121.  Final  English  Formula 76 

122.  French  Formulas 77 

123.  Babinet's  Formula 77 

124.  Tables 78 

125.  Approximations 78 

CHAPTER  H. INSTRUMENTS. 

126.  Mountain  Barometers 79 

127.  The  Aneroid  Barometer 80 

128.  "  Boiling-point  Barometer  " •. 80 

129.  Accuracy  of  Barometric  Observations 81 

130.  Simultaneous  Observations . . . . 81 


PART  IV. 

TOPO  GRAPHT. 

INTRODUCTION. 

131.  Definition 82 

132.  Systems .82 

CHAPTER  I. BY   HORIZONTAL   CONTOUR-LINES. 

133.  General  Ideas 83 

134.  Plane  of  Eeference 84 

135.  Vertical  Distances  of  the  Horizontal  Sections 84 

136.  Methods  for  determining  Contour-lines 84 


fc  ANALYTICAL   TABLE   OF.  CONTENTS. 

First  Method. 
U*T.  PAGE 

137.  General  Method 84 

138.  On  a  Long,  Narrow  Strip 85 

139.  On  a  Broad  §urface 85 

140.  Surveying  the  Contour-lines 85 

141.  Contouring  with  the  Plane-table 86 

Second  Method. 

142.  General  Nature 86 

143.  Irregular  Ground 86 

144.  On  a  Single  Hill. '. 87 

145.  Extensive  Topographical  Survey .-. .  87 

146.  Interpolation 88 

147.  Interpolating  with  the  Sector 88 

148.  Ridges  and  Thalwegs 89 

149.  Forms  of  Ground 90 

150.  Sketching  Ground  by  Contours 91 

1$1.  Ambiguity 91 

152.  Conventionalities 92 

153.  Applications  of  Contour-lines 92 

154.  Sections  by  Oblique  Planes 92 

CHAPTER  II. — BY   LINES   OF   GREATEST   SLOPE. 

155.  Their  Direction 93 

156.  Sketching  Ground  by  this  System 93 

157..  Details  of  Hatchings 93 

CHAPTER  HI. — SHADES   FROM   OBLIQUE   AND  VERTICAL   LIGHT. 

158.  Degree  of  Shade ..'. 94 

159.  Shades  l*y  Tints. . . 94 

160.  Shades  by  Contour-lines 95 

161.  Shades  by  Lines  of  Greatest  Slope 95 

162.  The  French  Method 95 

163.  Lehmann's  Method 95 

164.  Diapason  of  Tints. 97 

165.  Shades  Produced  by  Oblique  Light 97 

CHAPTER  IV. CONVENTIONAL   SIGNS. 

166.  Signs  for  Natural  Surface 98 

167.  Signs  for  Vegetation 98 

168.  Signs  for  Water 99 


ANALYTICAL  TABLE  OF  CONTENTS.  xi 

ART.  PAGE 

169.  Colored  Topography 100 

170.  Signs  for  Miscellaneous  Objects 101 

171.  Scales  . .  .103 


PART  V. 

UNDERGROUND    OR   MINING    SURVEYING. 

172.  Objects 105 

CHAPTER  I. SURVEYING  AND  LEVELLING  OLD  LINES. 

173.  First  Object 105 

174.  The  Old  Method * 106 

175.  The  New  Method 108 

176.  The  Mining  Transit 109 

177.  Mapping 109 

CHAPTER   H. LOCATING   NEW   LINES. 

178.  Second  Object -  ..  110 

179.  When  the  Mine  is  entered  by  an  Adit 110 

180.  When  the  Mine  is  entered  by  a  Shaft Ill 

181.  To  Dispense  with  the  Magnetic  Needle Ill 

182.  Keducing  Several  Courses  to  One 112 

183.  Third  Object 113 

184.  Problems .113 


PART  VL 

THE  SEXTANT  AND    OTSER  REFLECTING   INSTRUMENTS. 

CHAPTER   I. — THE   INSTRUMENTS. 

185.  Principle U5 

186.  Description  of  the  Sextant .  117 

187.  The  Box  Sextant 118 

188.  The  Keflecting  Circle '  us 

189.  Adjustments  of  the  Sextant 118 

190.  How  to  Observe 120 

191.  Parallax  of  the  Sextant ..  .  120 


xii  ANALYTICAL  TABLE  OF  CONTENTS. 

CHAPTER  H. — THE  PRACTICE. 

ABT.  PAGE 

192.  To  Set  Out  Perpendiculars 121 

193.  The  Optical  Square 121 

194.  To  Measure  a  Line,  one  end  being  inaccessible 122 

195.  Otherwise 124 

196.  To  Measure  a  Line  when  both  ends  are  inaccessible 124 

197.  Obstacles 124 

198.  To  Measure  Heights 124 

199.  Artificial  Horizon 125 

200.  The  Sun 126 

201.  Very  Small  Altitudes  and  Depressions 126 

202.  To  Measure  Slopes 127 

203.  Oblique  Angles 128 

204.  Advantages  of  the  Sextant 129 


•     PART  VII. 

MARITIME   OB  HYDRO GRAPHICAL  SURVEYING. 

205.  Object ;  131 

CHAPTER   I. THE   SHORE   LINE. 

206.  The  High-water  Line 131 

207.  The  Low-water  Line 132 

208.  Measuring  the  Base 132 

• 

CHAPTER   H. — SOUNDINGS. 

209.  In  Narrow  Water 133 

210.  Finding  the  Position  of  a  Boat  on  a  Sea-coast 134 

211.  From  the  Shore 134 

212.  From  the  Boat,  with  a  Compass 134 

213.  From  the  Boat,  with  a  Sextant 134 

214.  Trilinear  Surveying 135 

215.  Problem  of  the  Three  Points 135 

216.  Instrumental  Solution 137 

217.  Analytical  Solution 137 

218.  Between  Stations 138 

219.  The  Sounding-line ' : 139 


ANALYTICAL  TABLE   OF   CONTENTS.  x{{{ 

CHAPTER  m. — TIDE-WATERS. 

ART.  PAGE 

220.  Tides 140 

221.  Difference  on  Atlantic  and  Pacific  Coast 140 

222.  Mean  Level  of  .the  Sea 141 

223.  High  and  Low  Water 141 

224.  "  Establishment "  of  a  Place 141 

225.  Tide  Gauges » 141 

226.  Tide  Tables ^ 142 

227.  Gauges  in  Bends *. . . 144 

228.  Beacons  and  Buoys 144 


CHAPTER   VI. — THE   CHART. 

229.  Methods  of  Fixing  Points  on  the  Chart 145 

230.  Conventional  Signs 146 


PART  VIII. 

SPHERICAL  SURVEYING,    OR   GEODESY. 

CHAPTER  I. — THE   FIELD-WORK. 

231.  Nature 147 

232.  Triangular  Surveying , 147 

233.  Outline  of  Operations 148 

234.  Measuring  the  Base 148 

235.  Corrections  of  the  Base 152 

236.  Eeducing  the  Base  to  the  Level  of  the  Sea 152 

237.  A  Broken  Base 153 

238.  Base  of  Verification 153 

239.  Choice  of  Stations 154 

240.  Signals 157 

241.  Observations  of  the  Angles 159 

242.  Keduction  to  the  Centre 160 

243.  The  Angles 162 

244.  "Spherical  Excess" 162 

245.  Correction  of  the  Angles ' 164 

246.  Interior  Filling-up .164 


ANALYTICAL  TABLE  OF   CONTENTS. 


CHAPTER   H.  -  CALCULATING   THE   SIDES   OF   THE   TRIANGLES, 

ART.  PAGE 

247.  Methods  .........................    .......................  165 

248.  Delambre's  Method  .......................................   165 

249.  Legendre's  Method  ........................................  166 

250.  Coordinates  of  the  Points  ..................................  166 

251.  Problem  1  ...............................................  167 

252.  Second  Solution  ....................  .  ..................  .  168 

253.  Problem  II  ...........................  .  ..................  .168 

254.  Lee's  Formulas.  .  .170 


LEVELLING,  TOPOGEAPHY,  AND  HIGHER 
SURVEYING. 


INTRODUCTION. 

(1.)  Levelling  in  General.  A  level  surface  is  one  which  is 
everywhere  perpendicular  to  the  direction  of  gravity,  as  indi- 
cated by  a  plumb-line,  etc. ;  and,  consequently,  parallel  to  the 
surface  of  standing  water.  It  is,  therefore,  spherical  (more 
precisely,  spheroidal),  but,  for  a  small  extent,  may  be  consid- 
ered as  plane.  Any  line  lying  in  it  is  a  level  line. 

A  vertical  line  is  one  which  coincides  with  the  direction 
of  gravity. 

The  height  of  a  point  is  its  distance  from  a  given  level 
surface,  measured  perpendicularly  to  that  surface,  and  there- 
fore in  a  vertical  line. 

LEVELLING  is  the  art  of  determining  the  difference  of  the 
heights  of  two  or  more  points. 

To  obtain  a  level  surface  or  line,  usually  the  latter,  is  the 
first  thing  required  in  levelling. 

When  this  has  been  obtained,  by  any  of  the  methods  to  be 
hereafter  described,  the  desired  height  of  a  point  may  be  de- 
termined directly  or  indirectly. 

(2.)  Direct  Levelling.  In  this  method  of  levelling,  a  level 
line  is  so  directed  and  prolonged,  either  actually  or  visually, 
as  to  pass  exactly  over  or  under  the  point  in  question  (i.  e., 
BO  as  to  be  in  the  same  vertical  plane  with  it),  and  the  height 


2  LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

(or  depth)  of  the  point  above  (or  below)  this  level  line  is 
measured  by  a  vertical  rod,  or  by  some  similar  means.  The 
height  of  any  other  point  being  determined  in  the  same  man- 
ner, the  difference  of  the  two  will  be  the  height  of  one  of  the 
points  above  the  other.  So  on,  for  any  number  of  points. 

DIRECT  LEVELLING  is  the  method  most  commonly  em- 
ployed. It  will  form  Part  I.  of  this  volume. 

(3.)  Indirect  Levelling.  In  this  method  of  levelling  the  de- 
sired height  is  obtained  by  calculation  from  certain  coordinate 
measured  lines  or  angles,  which  fix  the  place  of  the  point. 

Thus,  the  horizontal  distance  from  any  point  to  a  tree 
being  known,  and  also  the  angle  with  the  horizon  made  by  a 
straight  line  passing  from  the  point  to  the  top  of  the  tree,  its 
height  above  the  point  can  be  readily  calculated.  This  is  the 
most  simple  and  most  usual  form  of  this  method,  though 
many  others  may  be  employed.  i 

INDIRECT  LEVELLING  will  be  developed  in  Part  II. 

(4.)  Barometric  Levelling,  This  determines  the  difference  of 
the  heights  of  two  points  by  the  difference  of  the  weights  of 
the  portions  of  the  atmosphere  which  are  above  each  of  them, 
.as  indicated  by  a  barometer.  It  is  explained  in  Part  III. 

(5.)  Topography.  " Surveying"  determines. the  position  of 
one  point  with  reference  to  another,  supposing  them  both  to 
be  situated  in  (or  reduced  to)  the  same  level  plane.  "  Level- 
ling "  determines  how  much  the  point  in  question  is  above  or 
below  some  other  level  plane.  Both  of  these  combined  de- 
termine where  the  point  is  "  in  space ; "  that  is,  where  it  is  in 
reference  to  some  known  point ;  both  horizontally,  i.  e.,  how 
far  it  is  in  front  or  behind,  to  right  or  to  left,  etc.;  and  ver- 
tically, i.  e.,  how  far  it  is  above  or  below. 

The  position  of  a  point  in  its  own  level  plane  is  usually 
determined  in  "  Surveying "  by  a  pair  of  coordinates — lines 
or  angles.  [See  L.  S.,  (2),  etc.]  l  Then,  its  vertical  distance 

1  L.  S.  will,  for  brevity,  be  used  to  denote  the  Author's  "  Treatise  on  Land 
Surveying,"  and  the  Xo.  of  the  article  referred  to  will  be  enclosed  in  (  ). 


INTRODUCTION.  3 

above  or  below  a  known  level  plane  (i.  e.,  its  height  or  depth) 
being  determined  by  "  Levelling,"  becomes  a  "  third  .  coordi- 
nate," which  fixes  the  place  of  the  point. 

The  application  of  such  combinations  of  Surveying  and 
Levelling  to  determine  the  positions,  in  horizontal  projection, 
and  also  the  heights  of  the  inequalities  of  a  limited  portion 
of  the  surface  of  the  earth  (its  hills  and  hollows,  ridges  and 
valleys,  etc.),  is  called  TOPOGRAPHY.  Topographical  Mapping 
represents  these  inequalities  on  paper.  Topography  on  a 
larger  scale  becomes  Geography,  properly  so  called.  TOPOG- 
RAPHY occupies  Part  IY. 

(6.)  Special  Objects  and  Difficulties.  The  preceding  methods 
are  sufficient  for  the  complete  determination  of  all  the  features 
of  the  earth's  surface;  but  certain  operations  in  particular 
places  require  special  methods. 

(7.)  Operations  beneath  the  surface  (for  tunnelling,  mining, 
etc.)  being  in  darkness,  and  not  easily  connected  with  the 
above-ground  work,  involve  some  novel  problems,  and  will, 
therefore,  be  treated  separately  in  Part  Y.,  as  UNDERGROUND 
SURVEYING  AND  LEVELLING. 

(8.)  So  too,  operations  on  the  water,  because  of  the  want  of 
steadiness  in  positions  on  its  surface,  require  peculiar  methods, 
and  constitute  another  modification ;  described  in  Part  YIL, 
as  WATER  SURVEYING  AND  LEVELLING. 

(9.)  REFLECTING  INSTRUMENTS,  such  as  the  sextant,  being 
chiefly  used  in  the  above  situation,  are  treated  of  in  the  pre- 
ceding Part  YI. 

(10.)  Spherical  Surveying,  "When  a  great  extent  ot  country 
is  comprised  in  a  survey,  the  surface  of  the  earth  can  no  longer 
be  considered  as  plane,  but  its  curvature  must  be  taken  into 
account.  Then  SPHERICAL  SURVEYING,  or  Geodesy,  must  be 
employed ;  and,  instead  of  the  straight  lines  and  plane  angles 


4:  LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

which  are  the  coordinates  of  Plane  Surveying,  arcs  of  circles 
and  spherical  angles  must  be  used.     It  forms  Fart  YIII. 

"When  still  greater  extents  are  to  be  surveyed,  the  methods 
of  spherical  surveying  must  be  modified  in  accordance  with 
the  true  spheroidal  form  of  the  earth. 

(11.)  Location,  The  name  Surveying  is  often  made  to  in- 
clude a  mode  of  operation  which  is  precisely  its  converse. 

Surveying,  properly  so  called,  determines  and  records  the 
relative  positions  of  points  as  they  really  are. 

The  converse  operations  have  for  their  objects  to  fix  the 
places  of  points  where  they  are  desired  to  be. 

The  term  LOCATION  may  be  extended  beyond  its  usual 
meaning  so  as  to  embrace  all  such  operations. 

In  laying  out  land,  parting  off  portions  of  it,  and  dividing 
it  up,  the  desired  lines  are  not  surveyed,  but  located. 

In  the  United  States  Public  Land  Surveying,  the  work  is. 
almost  entirely  Location. 

The  determination  of  the  lines  of  roads,  their  curves,  ete., 
is  especially  Location. 

The  finding  and  pursuing  a  given  course  at  sea  (in  Navi- 
gation) is  only  another  form  of  it. 

We  shall  find  many  applications  of  this  distinction  be- 
tween Surveying  and  Location.  A  similar  one  occurs  in 
Levelling.  It  should  be  carefully  kept  in  mind  both  in  "  Sur- 
veying" and  in  "Levelling." 


PART  I. 
DIRECT    LEVELLING. 


CHAPTER  I. 

GENERAL      PRINCIPLES. 

(12.)  Levelling  Instruments.  The  instruments  employed  to 
obtain  a  level  line  may  be  arranged  in  three  classes,  depending 
on  these  three  principles : 

1.  That  a  line  perpendicular  to  a  vertical  line  is  a  hori- 
zontal or  level  line. 

2.  That  the  surface  of  a  liquid  in  repose  is  horizontal. 

3.  That  a  bubble  of  air,  confined  in  a  vessel  otherwise  full 
of  a  liquid,  will  rise  to  the  highest  point  of  that  liquid. 

They  will  be  described  in  the  following  three  chapters. 

(13.)  Methods  of  Operation.  When  a  level- line  has  been  ob- 
tained, by  any  means,  'the  difference  of  heights  of  any  two 
points  may  be  found  by  either  of  these  two  methods : 

FIG.  1. 


First  Method. — Set  the  levelling  instrument  over  one  of 
the  points,  as  A,  in  Fig.  1.     Measure  the  height  of  the  level 


6 


LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


FIG.  2. 


"' 


0 


line  above  the  point.  Then  direct  this  line  to  a  rod  held  on 
the  other  point,  and  note  the  reading.  The  difference  of  the 
two  measurements  at  A  and  B  will  be  the  difference  of  their 
heights. 

Second  Method.     Let  A  and  B,  Fig.  2,  represent  the  two 
points.     Set  the  instrument  on  any  spot  from  which  both  the 

points  can  be  seen,  and 
at  such  a  height  that 
the  level  line  will  pass 
above  the  highest  one. 
Sight  to  a  rod  held  at 
A,  and  note  the  read- 
ing. Then  turn  the 
instrument  toward  B, 
and  note  the  height  ob- 
served on  the  rod  held 
at  that  point.  The  difference  of  the  two  readings  will  be  the 
difference  of  the  heights  required.  The  absolute  height  of  the 
level  line  itself  is  a  matter  of  indifference. 

(14.)  Curvature.     The  level  line  given  by  an  instrument  is 
tangent  to  the  surface  of  the  earth.     Therefore,  the  line  of 
true  level  is  always  below  the  line  of  apparent  level.    In  Fig.  3, 
A  D  represents  the  line  of  apparent  level, 
and  A  B  the  line  of  true  level.     D  B  is 
the  correction  for  the  earth's  curvature. 
By  geometry  we  have : 

ADa  =  DB  x  (DB-f  2 BO). 

But  D  B,  being  very  small,  compared  with 
the  diameter  of  the  earth,  may  be  dropped 
fromjfne  quantity  in  the  parenthesis,  and 

weJnave : 

AD2 


Fio.  3. 


"2BO 

i.  e.,  the  correction  equals  the  square  of 
the  distance  divided  by  the  diameter  of  the  earth. 

^jj-Al 

M  A 


i^i 

. 


In" I 


PERPENDICULAR  LEVELS. 

The  difference  of  height  for  a  distance  of 


1  mile  = 


1 
7916 


5280  x  12 
7916 


=  8  inches. 


This  varies  as  the  square  of  the  distance.  The  effect,  if 
neglected,  is  to  make  distant  objects  appear  lower  than  they 
really  are. 

The  effect  is  destroyed  by  setting  the  instrument  midway 
between  the  two  points. 

(15.)  Refraction.  Rays  of  light  coming  through  the  air  are 
curved  downward.  The  effect  is,  to  make  objects  look  higher 
than  they  really  are.  Its  amount  is  about  ^  that  of  curvature, 
and  it  operates  inj,  contrary  direction. 

/2.,//i//    ,£-./•     '       \i~ft' 

/  ;d> , :  •*'    -  -X' 


CHAPTER  II. 


PERPENDICULAR    LEVELS. 


(16.)  Principle.  The  principle  upon  which  these  are  con- 
structed is,  that  a  line  perpendicular  to  the  direction  of  gravity 
is  -a  level  line. 

(17.)  Plumb-line  Levels.  The  A  level,  Fig.  4,  is  so  adjusted 
that  when  the  plumb-line  coincides  with  the  mark  on  the 

FIG.  5. 


cross-piece,  the  feet  of  the  level  shall  be  at  the  same  height. 
It  is  adjusted  by  reversion  thus :  Place  its  feet  on  any  two 
points.  Mark  on  the  cross-bar  the  place  of  the  plumb-line. 


8  LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


Turn  the  instrument  end  for  end,  resting  it  on  the  same 
points,  and  mark  the  new  place  of  the  plumb-line.  The  point 
midway  between  the  two  is  the  right  one. 

Another  form  is  shown  in  Fig.  5. 

The  above  forms  are  not  convenient  for  prolonging  a  level 
line.  To  do  this,  invert  the  preceding  form,  as  in  Fig.  6. 


FIG.  6. 


FIG.  7. 


r 


To  test  and  adjust  this,  sight  to  some  distant  point  nearly 
on  a  level,  and  mark  where  the  plumb-line  comes  to  on  the 
bottom  of  the  rod.     Turn  the  instru- 
-j—  ment  around  and  sifjht  again,  and  note 
the  place  of  the  plumb-line.     The  mid- 
way point  is  the  right  one. 

A  modification  of  the  last  form  is  to 
fasten  a  common  carpenter's  square  in 
a  slit  in  the  top  of  a  staff,  by  means  of 
a  screw,  and  then  tie  a  plumb-line,  at 
the  angle  so  that  it  may  hang  beside 
one  arm.  When  it  has  been  brought  to 
to  do  so,  by  turning  the  square,  then  the  other  arm  will  be 
level. 

(18.)  Reflecting  Levels.  In  these,  the  perpendicular  to  the 
direction  of  gravity  is  not  an  actual  line,  but  an  imaginary 
reflected  line. 

It  depends  on  the  optical  principle  that  a  ray  of  light 
which  meets  a  reflecting  plane  at  right  angles  is  reflected  back 
in  the  same  line. 

When  the  eye  sees  itself  in  a  plane  mirror,  the  imaginary 
line  which  passes  from  the  eye  to  its  image  is  perpendicular 
to  the  mirror.  Therefore,  if  the  mirror  be  vertical,  the  line 


PERPENDICULAR  LEVELS. 


5 

\^  y*V 

will  be  horizontal.     It  may  therefore  be  used  likeN^W^JIgf] 
line  of  sight  for  determining  points  at  the  same  height  asrBe^ 


FIG.  8. 


The  first  form,  Fig.  8  (Colonel  Burel's),  consists  of  a  rhomb 
of  lead,  of  about  2  inches  on  a  side,  and  1  inch  thick. 

One  side  (the  shaded  part  of  the  figure)  is  faced  with  a 
mirror.      The  right-hand  corner  of  the   rhomb 
is  cut  off,  as  seen  in  the  figure,  and  a  wire,  A  B 
is  stretched  across  the  mirror. 

To  use  this,  hold  up  the  instrument,  with 
the  mirror  opposite  the  eye,  by  the  string  D,  A, 
so  that  the  eye  seems  bisected  in  the  mirror 
by  the  wire  AB.  Then  glance  through  the 
opening  at  B,  and  any  point  in  the  line  of  the 
eye  and  wire  will  be  in  the  same  horizontal 
plane  with  them. 

The  correctness  of  the  instrument  may  be  verified  in  the 
following  manner :  Hold  up  the  instrument  before  any  plane 
surface,  as  a  wall,  and  determine  the  height  of  some  point,  as 
previously  directed.  Then,  without  changing  the  height  of 
the  instrument,  turn  it  half  around,  place  yourself  between  it 
and  the  wall,  and  note  the  point  of  the  wall  which  is  seen  in 
the  mirror  to  coincide  with  the  image  of  the  eye. 

If  the  two  points  on  the  wall  coincide,  the  instrument  is 
correct.  If  they  do  not,  the  mirror  does  not  hang  plumb,  and 
the  point  midway  between  the  two  is  the  true  one. 

The  instrument  is 
rectified,  or  made  to 
hang  plumb,  by  means 
of  the  pear  -  shaped 
piece  of  lead  seen  at- 
:  tached  to  the  lower 
corner  of  the  rhomb. 

The    second   form 
consists   of    a  hollow 
brass  cylinder,  with  an 
opening  at  the  upper  end,  as  seen  in  Fig.  9.     At  the  opening 
is  a  small  mirror,  whose  vertical  plane  makes  an  angle  with 


FIG.  9. 


10-        LEYELLIXG,  TOPOGRAPHY,  AXD   HIGHER  SURVEYIXG. 


Fio  10. 


the  vertical  plane  of  section  by  which  the  cylinder  was  cut  in 
forming  the  aperture.  The  edge  of  the  mirror  is  marked  thus 
(  x )  in  the  first  half  of  Fig.  9.  The  mirror  is  made  to  hang 
plumb  by  means  of  a  one-sided  weight  within  the  cylinder. 

This  is  used  by  setting  it  on  a  stake  driven  into  the  ground, 

or  by  holding  it  in  the  hand,  making  the  lower  edge  of  the 

opening  answer  the  same  purpose  as  the  wire  in  the  other  case. 

The  same  methods  of  verification  and  rectification  are  used 

as  with  the  first  form  of  the  instrument. 

The  instrument,  in  its  third 
form,  is  simply  a  small  steel 
cylinder,  4"  or  5"  long,  and  ¥ 
in  diameter,  highly  polished, 
—  and  suspended  from  the  centre 
of  one  end  by  a  fine  thread. 

To  use  this,  hold  it  up  by  the 
thread  with  one  hand,  and  with 
the  other  hand  hold  a  card  be- 
tween the  eye  and  instrument, 

using  the  upper  edge  of  the  card,  as  seen  reflected  in  the 
mirror,  the  same  as  the  wire  in  the  first  form. 

This  instrument  is  the  invention  of  M.  Cousinery. 


CHAPTER  III. 

WATER-LEVEL  8. 

(19.)  Continuous  Water-levels,  These  may  consist  of  a  chan- 
nel connecting  the  two  points,  and  filled  with  water ;  or  of  a 
tube,  usually  flexible,  with  the  ends  turned  up  and  extending 
from  one  point  to  the  other. 

By  measuring  up  or  down,  frorii  the  surface  of  the  water  at 
each  end,  the  relative  heights  of  the  two  points  may  be  de- 
termined. 


WATER-LEVELS. 


(20.)  Visual  Water-levels.  The  simplest  one  is  a  short  sur- 
face of  water  prolonged  by  sights  at  equal  distances  above  it, 
as  in  Fig.  11. 


FIG.  11. 


A  portable  form  is  a  tube  bent  up  at  each  end,  and  nearly 
filled  with  water.  The  surface  of  the  water  in  one  end  will 
always  be  at  the  same  height  as  that  in  the  other,  however 
the  position  of  the  tube  may  vary.  It  may  be  easily  con- 
structed with  a  tube  of  tin,  lead,  copper,  etc.,  by  bending  up, 
at  right  angles,  an  inch  or 

Jf  T?  J  J  FlQ'    12« 

two  of  each  end,  and  sup-    ^ 

porting    the  tube,   if  too 

flexible,  on  a  wooden  bar. 

In  these  ends,  cement  (with 

putty,    twine    dipped    in 

white-lead,  etc.)  thin  phials,  with  their  bottoms  broken  off, 

so  as  to  leave  a  free  communication  between  them.     Fill  the 

tube  and  the  phials,  nearly  to  their  top,  with  colored  water. 

Blue  vitriol  or  cochineal  may  be  used  for  coloring  it.     Cork 

their  mouths,  and  fit  the  instrument,  by  a  steady  but  flexible 

joint,  to  a  tripod. 

To  use  it,  set  it  in  the  desired  spot,  place  the  tube  by  eye 
nearly  level,  remove  the  corks,  and  the  surfaces  of  the  water 
in  the  two  phials  will  come  to  the  same  level.  Stand  about 
a  yard  behind  the  nearest  phial,  and  let  one  eye,  the  other 
being  closed,  glance  along  the  right-hand  side  of  one  phial, 
and  the  left-hand  side  of  the  other.  Raise  or  lower  the  head 
till  the  two  surfaces  seem  to  coincide,  and  this  line  of  sight? 
prolonged,  will  give  the  level  line  desired.  Sights  of  equal 
height,  floating  on  the  water,  and  rising  above  the  tops  of  the 
phials,  would  give  a  better-defined  line. 


12         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


CHAPTER  IY. 

AIE-BUBBLE     OE     SPIRIT     LEVELS. 

(21.)  The  "Spirit-level"  consists  essentially  of  a  curved 
glass  tube  nearly  filled  with  alcohol,  but  with  a  bubble  of  air 
FlG  13  left  within,  which  always  seeks 

the  highest  spot  in  the  tube, 
and  will  therefore,  by  its  move- 
ments, indicate  any  change  in 
the  position  of  the  tube.  When- 
ever the  bubble,  by  raising  or  lowering  one  end,  has  been 
brought  to  stand  between  two  marks  on  the  tube,  or,  in  case 
of  expansion  or  contraction,  to  extend  an  equal  distance  on 
either  side  of  them,  the  bottom  of  the  block  (if  the  tube  be  in 
one),  or  sights  at  each  end  of  the  tube,  previously  properly 
adjusted,  will  be  on  the  same  level  line.  It  may  be  placed 
on  a  board  fixed  to  the  top  of  a  staff  or  tripod. 

When,  instead  of  the  sights,  a~ telescope  is  made  parallel 
to  the  level,  and  various  contrivances  to  increase  its  delicacy 
and  accuracy  are  added,  the  instrument  becomes  the  engi- 
neer's spirit-level. 

The  upper  surface  of  the  tube  is  usually  the  arc  of  a  circle, 
and  when  we  speak  of  lines  parallel  to  a . "  level,"  we  mean 
parallel  to  the  tangent  of  this  arc  at  its  highest  point,  as  indi- 
cated by  the  middle  of  the  bubble. 

(22.)  Sensibility.  This*  is  estimated  by  the  distance  which 
the  bubble  moves  for  any  change  of  inclination.  It  is  directly 
proportional  to  the  radius  of  curvature  of  the  tube.  To  de- 
termine the  radius,  proceed  thus : 

Let  S  =  length  of  the  arc  over  which  the  bubble  moves  for 
an  inclination  of  1  second 


AIR-BUBBLE   OR   SPIRIT   LEVELS. 


13 


Let  R  =  its  radius  of  curvature. 

Then  S  :  2?rR  ::  1"  :  360°, 
whence  R  =  206265  x  S, 
R 


or  S  = 


1  uac  &40*&  * 


206265 


FIG.  14. 


S  may  be  found  by  trial,  the  level  being  attached  to  a 
finely-divided  vertical  cir- 
cle.    The  radius  may  also 
be    found    without    this, 
thus :  Bring  the  bubble  to 
centre,  and  sight  to  a  di-    ^-^ 
vided  rod.    Raise  or  lower   ' 
one  end  of  the  level,  and 
again  sight  to  rod.     Call 
the  difference  of  the  read- 
ings A,  the  distance  of  the 
rod  d,  and  the  space  which 

the  bubble  moved  S.     Then  we  have  two  approximately  sim- 
ilar triangles ;  whence   r  =  -j-  • 

Example.  At  100  feet  distance,  the  difference  of  readings 
was  0.02  foot,  and  the  bubble  moved  0.01  foot.  Then  the  ra- 

100  x  0.01 
dius  was  — JT-— 9 =  50  ft. 

The  sensibility  of  an  air-bubble  level  equals  that  of  a 
plumb-line  level  having  a  plumb-line  of  the  same  length  as 
the  radius  of  curvature. 

(23.)  Block-level     If  this   is  marked  by  the   maker,  and 
the  bubble  does  not  come   to  the 
centre,  when  turned  end  for  end, 
plane  or  grind  off  one  end  of  the 
bottom  until  it  does. 

Otherwise,  if  the  bubble-tube  is  capable  of  movement, 
raise  or  lower  one  end  of  it  until  it  will  verify,  bringing  the 


FIG.  15. 


LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


bubble  half-way  back  to  the  middle  by  this  means,  and  the 
other  half  by  raising  or  lowering  one  end  of  the  block ;  be- 
cause the  reversion  has  doubled  the  error. 
Kepeat  this,  if  necessary. 


FIG.  16. 


Circular  Level.  The  upper  surface  of  this  is  spherical. 
It  will  therefore  indicate  a  level  in  every  di- 
rection, instead  of  only  one,  as  does  the  pre- 
ceding. It  is  adjusted  like  the  last  one,  but 
in  two  directions,  at  right  angles  to  each 
other. 

(24.)  Level  with  Sights.     The  line  of  sight  is  made  parallel 
to  the  tangent  of  the  level.     It  may  be  tested  thus  : 

FIG.  17. 


— ^~^—-^ 


Bring  the  bubble  to  the  centre  of  the  tube  and  make  a 
mark,  in  the  line  of  sight,  as  far  off  as  can  be  seen.  Then 
turn  the  level  end  for  end,  and  sight  again.  If  the  bubble 
remains  in  the  same  place,  "  all  right."  If  not,  rectify  it  by 
altering  the  sights,  or  by  altering  the  marks  for  the  bubble  to 
come  to,  bringing  the  bubble  half-way  back,  and  trying  it 
again. 

(25.)  Hand  Reflected  Level.  This  consists  of  a  brass  tube, 
about  six  inches  long,  and  one  inch  in  diameter.  To  the 
inside  of  the  upper  portion  of  the  tube  is  attached  a  small 
level.  A  small  mirror  is  placed  at  an  angle  in  the  lower  side 
of  the  tube,  so  that  it  will  reflect  the  point  to  which  the  bub- 
ble must  come,  in  order  to  have  the  instrument  level,  to  the 
eye.  A  small  hole  at  one  end,  and  a  horizontal  cross-hair  at 


AIR-BUBBLE  OR  SPIRIT  LEVELS. 


15 


the  other,  gives  the  desired  level  line.     It  is  used  by  holding 
it  in  the  hand. 


FIG.  18. 


Fig.  18  is  a"n  approved  form,  made  by  Young,  of  Phila- 
delphia. The  improvement  consists  in  the  patent  "Locke 
sight,"  which  enables  the  near  cross-hair  to  be  distinctly  seen 
at  the  same  time  as  the  distant  object. 


FIG.  19. 


(26.)  The  Telescope  Level.  In  this  the  line  of  collimation 
of  the  telescope  corresponds  to  the  sights  of  Fig.  17,  and  is 
made  parallel  to  the  level ;  i.  e.,  this  line  is  so  adjusted  as  to 
be  horizontal  when  the  bubble  of  its  level  is  in  the  centre. 

There  are  many  different  forms  of  the  Telescope  Level,  of 
which  the  most  important  ones  will  now  be  given. 

NOTE. — The  level,  represented  in  Fig.  19,  and  described  in  the  following  arti- 
cles, and  the  Transit,  represented  in  Fig.  73,  and  described  in  Art.  (92),  are  made 
by  W.  &  L.  E.  Gurley,  of  Troy,  N.  Y.,  to  whom  the  editor  is  indebted  for  valuable 
information  respecting  the  construction  and  adjustments  of  the  instruments. 


16         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


(27.)  The  Y  Level.     This  is  so  named  from       a  FlG- 
the  shape  of  the  supports  of  the  telescope.     It 
is  the  variety  most  used  by  American  engi- 
neers. 

Fig.  19  represents  a  twenty-inch  Y  level 
of  the  usual  form.  The  telescope  is  held  in 
the  wyes  by  the  clips,  A  A,  which  are  fast- 
ened to  the  wyes  by  tapering  pins,  so  that 
the  telescope  can  be  clamped  in  any  position. 
The  milled-headed  screws  at  M  and  M  are 
used  to  move  the  object-glass  and  eye-piece  in 
and  out,  so  as  to  adjust  them  for  long  and 
short  sights,  and  for  short-sighted  and  long- 
sighted persons.  L  is  a  spirit-level ;  P  and  P 
are  parallel  plates;  C  is  the  clamp-screw, 
which  fastens  the  spindle  on  which  the  level 
bar,  B,  which  supports  the  wyes,  turns ;  T  is 
the  tangent  screw,  by  which  the  telescope  may 
be  slowly  turned  around  horizontally. 

(28.)  The  Telescope.  The  arrangement  of 
the  parts  of  the  telescope  is  shown  in  Fig.  20. 
O  is  the  object-glass,  by  which  an  image  of 
any  object,  toward  which  the  telescope  may 
be  directed,  is  formed  within  the  tube.  E  E  is 
the  eye-piece — a  combination  of  lenses,  so  ar- 
ranged as  to  magnify  the  small  image  formed 
by  the  object-glass.  The  cross-hairs  are  at  X. 
They  are  moved  by  means  of  the  screws  shown 
at  B  B.  A  A  are  screws  used  for  centring  the 
eye-piece.  C  C  are  screws  used  for  centring 
the  object-glass.  At  D  D  are  rings,  or  collars, 
of  exactly  the  same  diameter,  turned  very  truly, 
by  which  the  telescope  revolves  in  the  wyes. 

The  telescope  shown  in  the  figure  forms  the  image  erect. 
Other  combinations  of  lenses  are  used,  some  of  which  invert 
the  image  ;  but  the  one  here  shown  is  generally  preferred. 


AIR-BUBBLE   OR   SPIRIT    LEVELS. 


17 


FIG.  21. 


(29.)  The  Cross-hairs.  These  are  made  of  very  fine  pla- 
tinum wire  or  of  spider-threads.  They  are  attached  to  a 

short,  thick  tube,  placed 
within  the  telescope- 
tube,  through  which  pass 
loosely  four  screws  whose 
threads  enter  and  take 
hold  of  the  cross-hair 
ring,  as  shown  in  Fig. 
21. 

In  some  instruments, 
one  of  each  pair  of  op- 
posite screws  is  replaced 
by  a  spring;  and  the 
screws,  instead  of  being  capstan-headed,  and  moved  by  an 
"  adjusting  -pin,"  have  square  heads,  and  are  moved  by  a 
"  key,"  like  a  watch-key. 

The  line  of  collimation  (or  line  of  aim)  is  the  imaginary 
line  passing  through  the  intersection  of  the  cross-hairs  and  the 
optical  centre  of  the  object-glass. 

The  image  formed  by  the  object-glass  should  coincide  pre- 
cisely with  the  cross-hairs.  When  this  is  not  the  case,  there 
will  be  an  apparent  movement  of  the  cross-hairs,  about  the 
objects  sighted  to,  on  moving  the  eye.  of  the  observer.  This 
is  called  instrumental  parallax.  To  correct  it,  move  the  eye- 
piece out  or  in,  till  the  cross-hairs  are  sharply  defined  against 
any  white  object.  Then  move  the  object-glass  in  or  out,  till 
the  object  is  also  distinctly  seen.  The  image  is  now  formed 
where  the  cross-hairs  are,  and  no  movement  of  the  eye  will 
cause  any  apparent  motion  of  the  cross-hairs. 

(30.)  The  Level.  This  consists  of  a  thick  glass  tube,  slightly 
curved  upward,  and  so  nearly  filled  with  alcohol  that  only  a 
small  bubble  of  air  remains  in  the  tube.  This  always  rises  to 
the  highest  part.  The  brass  case,  in  which  this  is  enclosed,  is 
attached  to  the  under  side  of  the  telescope,  and  is  furnished 
with  the  means  of  moving,  at  one  end  vertically,  and  at  the 
2 


18         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

other,  horizontally.  Over  the  aperture,  in  the  case,  through 
which  the  bubble-phial  is  seen,  is  a  graduated  level-scale,  num- 
bered each  way  from  zero  at  the  centre. 

(31.)  Supports.  The  wyes  in  which  the  telescope  rests,  are 
supported  by  the  level-bar,  B,  and  fastened  to  it  by  two  nuts 
at  each  end  (one  above  and  one  below  the  bar),  which  may  be 
moved  with  an  adjusting-pin.  The  use  of  these  nuts  will  be 
explained  under  "  Adjustments."  Attached  to  the  centre  of 
the  level-bar  is  a  steel  spindle,  made  so  as  to  turn  smoothly 
and  firmly  in  a  hollow  cylinder  of  bell-metal ;  this,  again,  is 
fitted  to  the  main  socket  of  the  upper  parallel  plate. 

(32.)  Parallel  Plates.  It  is  by  the  aid  of  these  that  the 
instrument  is  levelled.  The  plates  are  united  by  a  ball-and- 
socket  joint,  and  are  held  apart  by  the  four  plate-screws, 
Q  Q  Q  Q,  which  pass  through  the  upper  one,  and  press  against 
the  lower  one. 

To  level  the  instrument,  turn  the  telescope  till  it  is  brought 
over  a  pair  of  opposite  parallel  plate-screws.  Then  turn  the 
pair  of  screws,  to  which  the  telescope  has  been  made  parallel, 
equally  in  opposite  directions,  screwing  one  in  and  the  other 
out,  till  the  bubble  is  brought  to  the  centre.  Then  turn  the 
telescope  so  as  to  bring  it  over  the  other  pair  of  opposite 
screws,  and  bring  the  bubble  to  the  centre,  as  before. 

Repeat  the  operation,  as  moving  one  pair  of  screws  may 
affect  the  other. 

Sometimes  one  of  each  pair  of  opposite  screws  is  replaced 
by  a  strong  spring,  and  in  some  instruments  only  three  screws 
are  used. 

The  lower  plate  is  screwed  on  to  the  tripod-head.  For 
tripods,  see  Article  (45). 

(33.)  Adjustments.  The  line  of  collimation  of  the  telescope 
should  be  horizontal  when  the  bubble  is  in  the  centre  of  the 
tube ;  which  will  be  the  case  when  this  line  is  parallel  to  the 
plane  of  the  level.  But  both  this  line  and  this  plane  are 


AIR-BUBBLE   OR   SPIRIT  LEVELS. 

imaginary,  and  cannot  be  compared  together  directly, 
are  therefore  compared  indirectly.  The  line  of  collimation  is 
made  parallel  to  the  bottom  of  the  collars,  and  the  plane  of 
the  level  is  then  made  parallel  to  them. 

(34.)  First  Adjustment.  To  make  the  line  of  collimation 
parallel  to  the  bottoms  of  the  collars. 

Sight  to  some  well-defined  point,  as  far  off  as  it  can  be  dis- 
tinctly seen.  Then  revolve  the-telescope  half  around  in  its  sup- 
ports ;  i.  e.,  turn  it  upside  down.  If  the  line  of  collimation  was 
not  in  the  imaginary  axis  of  the  rings,  or  collars,  on  which  the 
telescope  rests,  it  will  now  no  longer  bisect  the  object  sighted  to. 
Thus,  if  the  horizontal  hair  was  too  high,  as  in  Fig.  22,  this  line 


FIG. 


of  collimation  would  point  at  first  to  A,  and,  after  being  turned 
over,,  it  would  point  to  B.  The  error  is  doubled  by  the  rever- 
sion, and  it  should  point  to  C,  midway  between  A  and  B. 
Make  it  do  so,  by  unscrewing  the  upper  capstan-headed  screw, 
and  screwing  in  the  lower  one,  till  the  horizontal  hair  is 
brought  half-way  back  to  the  point.  Remember  that,  in  an 
erecting  telescope,  the  cross-hairs  are  reversed,  and  vice  versa. 
Bring  it  the  rest  of  the  way  by  means  of  the  parallel  plate- 
screws.  Then  revolve  it  in  the  wyes  back  to  its  original  po-  ' 
sition,  and  see  if  the  intersection  of  the  cross-hairs  now  bisects 
the  point,  as  it  should.  If  not,  again  revolve,  and  repeat  the 
operation  till  it  is  perfected.  If  the  vertical  hair  passes  to 
the  right  or  to  the  left  of  the  point  when  the  telescope  is 
turned  half  around,  it  must  be  adjusted  in  the  same  manner 
by  the  other  pair  of  cross-hair  screws.  One  of  these  adjust- 
ments may  disturb  the  other,  and  they  should  be  repeated 
alternately.  When  they  are  perfected,  the  intersection  of  the 
cross-hairs,  when  once  fixed  on -a  point,  will  not  move  from  it 

§ 


20        LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

when  the  telescope  is  revolved  in  its  supports.  This  double 
operation  is  called  adjusting  the  line  of  collimation. 

It  has  now  been  brought  into  the  centre  line,  or  axis,  of  the 
collars,  and  is  therefore  parallel  to  their  bottoms,  or  the  points 
on  which  they  rest,  if  they  are  of  equal  diameters.  We  have 
to  assume  this  as  having  been  effected  by  the  maker. 

In  making  this  adjustment,  the  level  should  be  clamped, 
but  need  not  be  levelled. 


(35.)  Second  Adjustment,  To  make  the  bottoms  of  the  col- 
lars parallel  to  the  plane  of  the  level  •  i.  e.,  to  insure  their 
being  horizontal  when  the  bubble  is  in  the  centre. 

Clamp  the  instrument,  and  bring  the  bubble  to  the  centre 
by  the  parallel  plate-screws.  Take  the  telescope  out  of  the 
wyes,  and  turn  it  end  for  end.  If  the  bubble  returns  to  the 
.centre,  "  all  right."  If  not,  rectify  it,  by  bringing  the  bubble 
half-way  back,  by  means  of  the  nuts  which  are  above  and 
below  one  end  of  the  bubble-tube,  and  which  work  on  a  screw. 
Bring  it  the  rest  of  the  way  by  the  plate-screws,  and  again 
turn  end  for  end.  Repeat  the  operation,  if  necessary. 

*  If,  in  revolving  the  telescope  (as  in  the  first  adjustment), 
the  bubble  runs  toward  either  end,  it  must  be  adjusted  side- 
ways, by  means  of  two  screws  which  press  horizontally  against 
the  other  end  of  the  bubble-tube.  This  part  of  the  adjustment 
may  derange  the  preceding  part,  which  must,  therefore,  be 
tried  again. 

(36.)  Third  Adjustment.  To  cause  the  bubble  to  remain  in 
the  centre  of  the  tube  when  the  telescope  is  turned  around  hor- 
izontally. 

To  verify  this,  bring  the  bubble  to  the  centre  of  the  tube, 
and  then  turn  the  telescope  half-way  around  horizontally.  If 
the  bubble  does  not  remain  in  the  centre,  adjust  it  by  bringing 
it  half-way  back  by  means  of  the  nuts  at  the  end  of  the  level- 
bar.  Test  it  by  bringing  it  the  rest  of  the  way  back  by  the 
parallel  plate-screws,  and  again  turning  half-way  around. 

The  cause  of  the  difficulty -is,  that  the  plane  of  the  level  is 


AIR-BUBBLE   OR   SPIRIT    LEVELS.  21 

not  perpendicular  to  the  axis  about  which  it  turns,  and  that 
this  axis  is  not  vertical.  The  above  operations  correct  both 
these  faults. 

This  adjustment  is  mainly  for  convenience,  and  not  for 
accuracy,  except  in  a  very  small  degree. 

Some  instruments  have  no  means  of  making  the  third  ad- 
justment. They  must  be  treated  thus : 

Use  the  screws  at  the  end  of  the  bubble-tube,  to  cause  the 
bubble  to  remain  in  the  centre  when  the  level  is  turned  around 
horizontally.  Tken  make  the  line  of  collimation  parallel  to 
the  level  by  the  method  given  in  Art.  (38),  by  raising  or  low- 
ering the  cross-hairs. 

(37.)  The  operations  of  centring  the  eye-piece  and  object- 
glass  should  precede  the  first  three  which  we  have  just  ex- 
plained. 

Centring  the  Object-glass.  After  adjusting  the  line  of 
collimation  for  a  distant  object  (as  explained  in  the  "  First 
Adjustment,"  Art.  (34),  move  out  the  slide,  which  carries  the 
object-glass,  until  a  point  ten  or  fifteen  feet  distant  can  be 
distinctly  seen.  Then  turn  the  telescope  half  over,  as  befere, 
and  see  if  the  intersection  of  the  cross-hairs  bisects  the  point. 
If  not,  bring  it  half-way  back  by  the  screws  C  C,  Fig.  20, 
moving  only  one  pair  of  screws  at  a  time.  Repeat  the  opera- 
tion for  a  distant  point,  and  then  again  for  a  near  one,  if 
necessary.  We  have  now  adjusted  the  line  of  collimation  for 
long  and  short  sights,  and  may  assume  it  to  be  in  adjustment 
for  intermediate  ones,  since  the  bearings  of  the  slides  are  sup- 
posed to  be  true,  and  their  planes  parallel  to  each  other. 

Centring  the  Eye-apiece.  This  is  to  enable  the  observer 
to  see  the  intersection  of  the  cross-hairs  precisely  in  the  centre 
of  the  field  of  view  of  the  eye-piece.  It  is  adjusted  by  means 
of  four  screws,  two  of  which  are  shown  at  A,  A. 

These  operations  are  performed  by  the  maker  so  perma- 
nently as  to  need  no  further  attention  from  the  engineer,  and  the 
heads  of  the  screws,  by  which  these  adjustments  are  made,  are 
covered  by  a  thin  ring  which  protects  them  from  disturbance. 


22         LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

(38.)  Adjustment  by  setting  between* two  points,  or  the 
"  Peg  Method"  Drive  two  pegs  several  hundred  feet  apart, 
and  set  the  instrument  midway  between  them.  Level,  and 
sight  to  the  rod  held  on  each  peg.  The  difference  of  the  read- 
ings will  be  the  true  difference  of  the  heights  of  the  pegs ;  no 
matter  how  much  the  level  may  be  out  of  adjustment. 

Then  set  the  level  over  one  peg,  and  sight  to  the  rod  at 
the  other.  Measure  the  height  of  the  cross-hairs  above  the 
first  peg.  The  difference  of  this  and  the  reading  on  the  rod 
should  equal  the  difference  of  the  heights  of  the  two  points,  as 
previously  determined.  If  it  does  not,  set  the  target  to  the 
sum  or  difference  of  the  height  of  the  cross-hairs  above  the 
first  peg,  and  the  true  difference  of  height  of  the  points,  ac- 
cording as  the  first  point  is  higher  or  lower  than  the  second, 
and  hold  the  rod  on  the  second  point.  Sight  to  it,  and  raise 
or  lower  one  end  of  the  bubble-tube  until  the  horizontal  cross- 
hair does  bisect  the  target  when  the  bubble  is  in  the  centre. 
Then  perform  the  "  third  adjustment." 

Instead  of  setting  over  one  peg,  it  is  generally  more  con- 
venient to  set  near  to  it,  and  sight  to  a  rod  held  on  it, .and  use 
this  reading,  instead,  of  the  measured  height  of  the  cross-hairs. 


, 

2994 

2.301  < 

'            L— 

>Z  398 

\  .705 

/ 

fczSz~i 

-.693 

X.  B.  This  verification  should  always  be  used  for  every 
level,  even  after  the  three  usual  adjustments-  have  been  made ; 
for  it  is  independent  of  the  equality  of  the  collars. 

In  running  a  long  line  of  levels,  let  the  last  sight  at  night 
be  taken  midway  between  the  last  two  "  turning-point " 
pegs,  and  in  the  morning  try  their  difference  by  setting  close 
to  the  last  one.  This  tests  the  level  every  day  with  very  little 
extra  labor. 


AIRJBUBBLE  OR  SPIRIT  LEVELS.  23 

(39.)  Verification  ~by  another  Telescope.  Set  up  and  level 
the  instrument,  and  bisect  the  target  on  a  distant  rod.  Then 
turn  the  telescope  half  around  horizontally,  and  bring  the 
bubble  to  the  centre,  if  disturbed.  Then  take  another  tele- 
scope, of  about  the  same  magnifying  power,  but  with  a  larger 
object-glass.  Hold  it  close  to  the  object-glass  of  the  level,  and 
look  through  the  level  telescope.  You  will  see  the  cross-hairs 
plainly,  and  by  the  side  of  your  telescope  you  will  see  the 
target.  If  the  cross-hairs  bisect  the  target,  "  all  right."  If 
not,  adjust  as  in  last  method.  If  the  second  telescope  be  not 
larger  than  that  of  the  level,  hold  it  to  one  side. 

(40.)  Egault's  Level.  In  this  level  the  bubble-tube  is  not 
connected  with  the  telescope.  It  is  used  thus : 

FlG  24  Level,  and  sight  as 

usual.  Then  turn  the 
telescope  upside  down, 
end  for  end,  and  half 
way  around  horizon- 
tally, and  sight  again. 
Half  the  sum  of  the 
two  readings  is  the 
correct  one,  no  matter 
how  much  the  instrument  is  out  of  adjustment  (assuming  the 
collars  to  be  of  equal  size) ;  for  the  errors  then  cancel  each 
other.  This  is  the  one  used  principally  in  France. 

The  rod  used  with  it  is  marked  with  numbers  only  half  the 
real  heights  above  its  bottom.  Then  the  sum  of  the  readings 
is  the  true  one.  Thus  the  rod  itself  takes  the  mean  of  the 
readings. 

FIG.  25. 


(41.)  Troughton's  Level,     in  this  the  bubble-tube  is  perma- 
nently fastened  in  the  top  of  the  telescope-tube.    It  is  adjusted 


LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


by  the  "  peg  method,"  or  some  similar  one,  the  cross-hair 
being  moved  up  or  down  until  the  observation  gives  the  true 
difference  of  height  of  the  pegs  when  the  bubble  is  in  the 
centre.  Then  make  the  "Third  Adjustment,"  by  means  of 
the  screws  under  the  telescope. 

(42.)  Gravatt's  Level,  or  the  "Dumpy  Level."  Its  diameter 
is  very  great,  thus  giv- 
ing more  light.  Its 
bubble  is  on  the  top, 
and  can  be  seen  in  a 
small  inclined  mirror, 
by  the  observer.  It  al- 
so has  a  cross-level. 


FIG.  26. 


(43.)  Kourdaloue's 
Level,  This  is  a  modifi- 
cation of  Egault's.  The 
telescope  carries  a  steel 
prism  near  each  end ; 
one  of  which  rests  on  a  knife-edge,  and  the  other  on  the 
spherical  top  of  an  adjusting-screw. 

(44.)  Lenoir's  Level.     In  this,  the  telescope  carries,  at  each 
end,  a  steel  block,  whose  upper  and  lower  faces  are  made  very 

FIG.  27. 


perfectly  parallel.     They  are  plsteed  on  a  brass  circle,  which 
is  made  level  by  reversing  a  level  placed  upon  the  telescope. 


RODS. 


25 


(45.) 

FIG.  28. 


FIG.  29. 


Tripods.  These  consist  of  three  legs,  shod  with  iron, 
and  connected  by  joints  at  the 
top.  There  are  many  different 
forms,  the  most  common  of  which 
is  given  in  Fig.  19.  Other  forms 
are  given  in  Figs.  26,  28,  and  29. 
Lightness  and  stiffness  are  the  de- 
sired qualities.  Of  the  two  rep- 
resented in  Figs.  28  and  29,  the 
first  has  the  advantage  of  being 
simple  and  cheap ;  and  the  second 
of  being  light  and  yet  strong. 

Stephenson's  tripod  has  a  ball- 
and-socket  joint  below  the  par- 
allel plates,  so  as  to  admit  of 
being  at  once  set  nearly  level  on 
very  steep  slopes. 


CHAPTEE   Y. 


FIG.  30 


FIG.  31. 


RODS. 

(46.)  THESE  should  be  made  of  light,  well-seasoned  wood. 
A  plumb  or  level  attached  to  them  will 
show  when  they  are  held  vertically.  To 
detect  whether  the  rod  leans  to  or  from  the 
instrument,  its  front  may  be  angular  or 
curved.  If  angular,  when  held  leaning  tow- 
ard the  instrument,  the  lines  of  division 
will  appear  as  in  Fig.  30.  When  leaning 
from  the  instrument,  they  will  appear  as  in 
Fig%31.  They  are  usually  "divided  to  feet, 
tenths,  and  hundredths. 


26          LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


(47.)  Target.  This  is  a  plate  of  iron  or  brass,  attached  to 
the  rod  in  such  a  way  that  it  may  be  moved  up  and  down  the 
rod  and  clamped  in  any  position.  The  face  of  the  target 
should  be  painted  of  such  a  pattern  that,  when  sighting  to  it, 
it  may  be  very  precisely  bisected  by  the  horizontal  cross-hair. 
Some  of  the  many  varieties  are  given  in  Figs.  32-40. 


FIG.  32. 


FIG.  33. 


FIG.  34. 


FIG.  35. 


FIG.  36. 


FIG.  37. 


FIG.  38.  FIG.  39. 


FIG.  40. 


o 


Those  represented  in  Figs.  32,  33,  and  34,  are  bad,  because 
the  cross-hair  may  be  above  or  below  the  middle  of  the  target 
by  its  full  thickness,  as  magnified  by  the  eye-piece  of  the  tel- 
escope, without  the  error  being  perceptible.  The  next  three, 
Figs.  35,  36,  and  37,  depend  upon  the  nicety  with  which  the 
eye  can  determine  if  a  line  bisects  an  angle.  Fig.  38  depends 
upon  the  accuracy  with  which  the  eye  can  bisect  a  space. 
Fig.  39  depends  upon  the  accuracy  with  which  the  eye  can 
bisect  a  circle.  Figs.  36,  37,  and  40,  are  the  best  forms  for 
use.  Red  and  white  are  the  best  colors. 

A  good  method  of  moving  the  target  on  a  long  rod,  is  by 
means  of  pulleys  a-t  the  ends  of  the  rod.  A  woollen  cord 


RODS. 


FIG.  41. 


should  be  used,  on  account  bf  its  being  least 
affected  by  moisture. 

(48.)  Vernier's.  L.  S.  [343-357].  The  target 
carries  a  Vernier,  by  which  smaller  spaces  may 
be  measured  than  those  into  which  the  rod 
is  divided.  It  may  be  placed  on  the  side  of  an 
aperture,  in  the  face  of  the  target,  through 
which  the  divisions  on  the  rod  can  be  seen ;  or 
carried  on  the  back  or  side  of  the  rod  by  the 
target-clamp. 

(49.)  The  New-York  Rod.  This  is  in  two 
pieces,  sliding  one  upon  the  other,  and  con- 
nected together  by  a  tongue.  It  is  graduated 
to  tenths  and  hundredths  of  a  foot,  and  can  be 
read  to  thousandths  by  the  Yernier.  Up  to  six 
feet  the  target  is  used  as  on  other  rods.  For 
greater  heights,  the  target  is  fixed  at  6J  ft.,  and 
the  back  part  of  the  rod,  which  carries  the  tar- 
get, is  shoved  up  (Fig.  41)  until  the  target  is 
bisected  by  the  cross-hairs.  Its  height  is  then 
read  off  on  the  side  of  the  rod ;  on  which  the 
numbers  run  downward,  and  on  which  is  a 
second  vernier,  which  gives  the 
precise  reading.  It  is  convenient 
for  its  portability,  but  apt  to  bind 
or  slip  in  sliding ;  i.  e.,  to  be  too 
tight  or  too  loose,  as  the  weather 
is  moist  or  dry. 


(50.)  The  Boston  Rod,  This  is  in 
two  parts  like  the  New  York  rod. 
The  target  is  rectangular  (Fig.  42), 
and  is  fastened  to  one  of  the  pieces  near  its  extremity.  For 
heights  less  than  six  feet,  the  rod  is  held  with  the  target-end 
down,  and  the  target  is  moved  up  by  sliding  up  the  piece 


28          LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

which  carries  it.  'For  heights  above  six  feet,  the  rod  is  turned 
end  for  end,  bringing  the  target-end  up,  and  then  sliding  up 
the  piece  which  carries  the  target. 

(51.)  Speaking-Rods.     These  are  rods  which  are  read  with- 
out targets,  the  divisions  and  subdivisions  being  painted  on 


FIG.  45. 


FIG.  43. 


Fio.  44. 


the  face  of  the  rod.  They  produce  great  saving  of  time  and 
increase  of  accuracy. 

In  one  form,  Fig.  43,  the  face  of  the  rod  is  divided  into 
tenths  of  feet,  and  smaller  divisions  estimated. 

In  Bourdaloue's  rod  the  divisions  are  each  4  centimetres 
(1.6  inches),  and  are  numbered  at  half  their  value.  He 
arranges  them  as  in  Fig.  44. 

GrwMs  Rod,  Fig.  45.  This  is.  divided  to  0.01  foot.  The 
upper  hundredth  of  each  tenth  extends  across  the  rod.  Each 
half-tenth  is  marked  by  a  dot.  Each  half-foot  by  two  dots. 
Every  other  tenth  is  numbered,  and  the  numbers  are  each 


RODS. 


29 


0.1  high.  It  is  in  three  parts,  which  slide  into  each  other  like 
a  telescope. 

Barlow's  Rod,  Fig.  46.  In  this  the  divisions  are  marked 
by  triangles,  each  0.02  ft.  high,  so  that  it  reads  to  hundredths, 
and  less  by  estimation.  This  is  based  on  the  power  the  eye 
has  in  bisecting  angles. 

Stephensorfs  Rod,  Fig.  47.  This  is  based  upon  the  prin- 
ciple of  the  Diagonal  Scale.  Each  tenth  is  bisected  by  a  hor- 
izontal line,  and  the  diagonals  enable  the  observer  to  read  to 
hundredths. 

Conybeare's  Rod,  Fig.  48.  It  reads  to  hundredths  of  a 
foot  by  means  of  the  cross-hair  bisecting  the  tops  and  bottoms 
and  angles  of  hexagons.  The  odd  tenths  are  made  white  and 
the  even  ones  black.  The  figures  are  placed  so  that  their 
centres  are  opposite  the  divisions  they  refer  to. 

Pembertorfs  Rod,  Fig.  49.     This  is  on  the  principle  of  9 


FIG.  47. 


FIG.  48. 


FIG.  46. 


FIG.  49. 


verniers  placed  side  by  side.  It  reads  to  hundredths,  which 
are  given  by  counting  up  from  the  dot  which  the  hair  bisects, 
to  the  dot  in  the  same  vertical  line  which  is  bisected  by  one 


30         LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

of  the  horizontal  lines  which  mark  the  tenths.     The  inventor 
claims  that  it  can  be  read  9  times  as  far-  as  Gravatt's. 

On  all  speaking-rods,  to  avoid  confounding  numbers,  such 
as  3  and  8,  they  may  be  marked  thus  : 

1  .  2  .  Ill  .  4  .  V  .  6  .  7  .  8  .  IX  .  X  .  11  .  XII. 

The  French,  who  go  by  tenths,  use  the  following  : 
I.2.T.4.V.6.7.8.  N.X. 

The  figures  are  sometimes  placed  with  their  tops  on  a 
level  with  the  tops  of  the  dimensions  they  mark,  e.  g.,  feet ; 
and  sometimes  with  their  middles  on  the  dividing  line. 


CHAPTEE  VI, 

') 

THE      PRACTICE. 

(52.)  Field  Routine:  or,  how  to  start  and  go  on. 

1.  The  rodman  holds  the  rod  on  the  starting-point ;  which 
may  be  a  peg,  a  door-sill,  or  other  "  bench-mark "  Art.  (60). 
He  stands  square  behind  his  rod,  and  holds  it  as  nearly  ver- 
tical as  possible. 

2.  The  leveller  sets .  up   the  instrument,  somewhere  in 
the  direction  in  which  he  is  going,  but  not  necessarily,  or 
usually,  in  the  precise  line.     He  then  levels  the  instrument 
by  the  parallel  plate-screws,  sights  to  the  rod,  and  notes  the 
reading,  whether  of  target  or  speaking-ro.d,  as  a  "  back-sight " 
(B.  S.)?  or  +  (plus)  sight ;  entering  it  in  the  proper  column 
of  one  of  the  tabular  forms  of  field-book,  given  in  the  follow- 
ing articles. 

3.  The  rodman  is  then  sent  ahead  about  as  far  as  he  was 
behind,  and  he  there  drives  a  "  level-peg  "  nearly  to  the  sur- 


THE  PRACTICE.  31 

face  of  the^  ground,  or  finds  a  hard,  well-defined  point,  and 
holds  the  rod  upon  it. 

4.  The  leveller  then  again  sights  to  the  rod,  and  notes 
the  reading  as  a  "  fore-sight "  (F.  S.),  or  —  (minus)  sight.    The 
difference  of  the  two  readings  is  the  difference  of  the  heights 
of  the  points. 

5.  He  then  takes  up  the  instrument,  goes  beyond  the  rod, 
any  convenient  distance,  sets  up  again,  and  proceeds  as  in 
paragraph  2 ;  and  so  on  for  any  number  of  points,  which  will 
form  a  series  of  pairs.     The  successive  observations  of  each 
pair  give  their  difference  of  heights,  and  the  combination  of 
all  these  gives  the  difference  of  heights  of  the  first  and  last 
points  of  the  series. 

6.  If  the  vertical  cross-hair  be  strictly  vertical,  it  will  de- 
termine whether  the  rod  leans  to  the  right  or  left.     To  know 
whether  the  cross-hair  is  vertical  or  not,  try  whether  it  coin- 
cides with  a  plumb-line ;  or  sight  to  some  fixed  point,  turn 
the  telescope  from  side  to  side  horizontally,  and  see  if  the  hor- 
'izontal  cross-hair  continues  to  cover  the  spot.     If  it  does  not, 
turn  the  telescope  around  in  the  wyes  till  it  does  ;  then  it  is 
truly  horizontal,  and  the  other  hair,  being  perpendicular  to  it, 
is  truly  vertical.     To  know  whether  the  rod  leans  forward  or 
backward,  have  the  rodman  move  it  from  and  to  himself.     If 
the  line  bisected  by  the  cross-hair  descends  in  both  motions, 
the  rod  was  vertical.     If  the  line  rises,  the  rod  was  leaning. 
The  lowest  reading  is  the  true  one. 

7.  .When  a  target  is  used,  signals  are  made  by  the  leveller 
with  the  hand,  "  up "  and  "  down,"  to   indicate   in   which 
direction  to  move  the  target.     Drawing  the  hand  to  the  side 
signifies  "  stop,"  and  both  hands  brought  together  above  the 
head  signifies  "  all  right."     The  rodman  should  move  the  tar- 
get fast  at  first,  and  slowly  after  having  passed  the  right  point. 
When  signalled  "  all  right,"  he  should  clamp  the  target  and 
show  again.     Then  call  out  the  reading  before  moving,  and 
show  it  to  the  leveller,  as  either  passes  the  other. 

8.  We  have  thus  far  supposed  that  only  the  difference  of 
heights  of  the  two  extreme  points  is  desired.     But  when  a 


32        LEVELLING,  TOPOGRAPHY,  AXD  HIGHER  SURVEYING. 

section  or  profile  of  the  ground  is  required,  the  rod  must  be 
held  and  observed,  at  each  change  of  slope  of  the  ground ;  or 
at  regular  distances ;  usually,  for  railroad  work,  at  every  hun- 
dred feet,  and  also  at  any  change  of  slope  between  those 
points. 

Any  number  of  points,  within  sight,  may  have  their  relative 
heights  determined  at  one  setting  of  the  level. 

The  names  back-sight  (B.  S.)  and  fore-sight  (F.  S.)  do  not 
necessarily  mean  sights  taken  looking  forward  or  backward 
(though  they  are  generally  so  for  turning-points),  but  the  first 
sight  taken,  after  setting  up  the  instrument,  is  a  B.  S.  or  + 
(plus)  sight,  and  all  following  ones,  taken  before  removing 
the  instrument,  are  F.  S.'s,  or  —  (minus)  sights.  The  full 
meaning  of  this  will  appear  in  considering  the  forms  of  field^ 
book. 

All  but  the  first  and  last  points  sighted  to  are  called  inter* 
mediate  points,  or  " intermediates"  The  last  point  sighted 
to  before  moving  the  instrument  is  called  a  turning-point  or 
changing-point. 

The  first  and  last  sights,  taken  at  any  one  setting  of  the 
instrument,  require  the  greatest  possible  accuracy.  The  in- 
termediate points  may  be  taken  only  to  the  nearest  tenth,  or 
hundredth  at  most ;  because  any  error  in  them  will  not  affect 
the  final  result,  but  only  the  height  of  that  single  point  at 
which  it  was  taken. 

Two  rodmen  are  often  used  to  save  the  time  of  the  leveller. 
Then  it  is  well  to  use  a  target-rod  for  the  "  turning-points," 
which  are  often  distant  and  need  most  precision,  and  a  speak- 
ing-rod for  the  intermediate  points.  Where  one  rod  is  used, 
the  rodman  should  keep  notes  of  the  readings  at  the  turning- 
points. 

(53.)  Field-notes,  The  beginner  may  sketch  the  heights 
and  distances  measured,  in  a  profile  or  side  view,  as  in  Fig. 
50.  But  when  the  observations  are  numerous,  they  should  be 
placed  in  one  of  the  tabular  forms  given  on  the  following 
pages. 


THE  PRACTICE. 
FIG.  50. 


33 


(54.)  First  Form  of  Field-book.  In  this,  the  names  of  the 
points,  or  "  Stations,"  whose  heights  are  demanded,  are  placed* 
in  the  first  column  ;  and  their  heights,  as  finally  ascertained,  in 
reference  to  the  first  point,  in  the  last  column.  The  heights 
above  the  starting-point  are  marked  -f ,  and  those  below  it  are 
marked  — .  The  back-sight  to  any  station  is  placed  on  the 
line  below  the  point  to  which  it  refers.  "When  a  back-sight 
exceeds  a  fore-sight,  their  difference  is  placed  in  the  column 
of  "  Bise ; "  when  it  is  less,  their  difference  is  a  "  Fall."  The 
following  table  represents  the  same  observations  as  the  last 
figure,  and  their  careful  comparison  will  explain  any  obscuri- 
ties in  either : 


Stations. 

Distances. 

Back-sights. 

Fore-sights. 

"Eise. 

Fall. 

To.  Heights. 

A 

0.00 

B 

100 

2.00 

6.00 

-  4.00 

—  4.00 

C 

60 

3.00 

4.00 

—  1.00 

-  5.00 

D 

40 

2.00 

1.00 

+  1.00 

—  4.00 

E 

70 

6.00 

1.00 

+  5.00 

+  i.oo 

F 

60 

2.00 

6.00 

—  4.00 

—  3.00 

15.00 

18.00 

—  3.00 

The  above  table  shows  that  B  is  4  feet  below  A  ;  that  C  is 
5  feet  below  'A ;  that  E  is  1  foot  above  A ;  and  so  on.  To  test 
the  calculations,  add  up  the  back-sights  and  fore-sights.  The 
difference  of  the  sums  should  equal  the  last  "  total  height." 


34:         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


An  objection  to  this  form  is  that  the  back-sights  come  on 
the  line  Mow  the  station  to  which  they  are  taken,  which  is 
embarrassing  to  a  beginner. 

When  "  intermediate  "  observations  are  taken,  the  "  fore- 
sights," taken  to  these  intermediate  points,  are  put  down  in 
their  proper  column,  and  are  also  set  down  in  the  column  of 
"  back-sights ; "  so  that  when  the  two  columns  are  added  up, 
any  error  in  these  intermediate  sights  (which  are  usually  not 
taken  very  accurately)  will  be  cancelled,  and  will  not  affect 
the  final  result.  The  effect  is  the  same  as  if,  after  the  fore- 
sight to  the  intermediate  point  had  been  taken,  the  instrument 
had  been  taken  up  and  set  down  again  at  precisely  the  same 
•height  as  before,  and  a  back-sight  had  then  been  taken  to  the 
same  point.  Hence,  in  this  form,  the  "  turning-points  "  are 
those  stations  which  have  different  back-sights  and  fore-sights, 
while  those  which  have  them  the  same  are  "  intermediates." 

The  following  figure  and  table  represent  the  s'ame  ground 
as  the  preceding  one,  but  with  only  two  settings  of  the  instru- 
ment. D  is  the  turning-point : 

FIG.  51. 


Stations. 

Distances. 

B.  S.  + 

F.  S.  - 

Rise. 

Fall. 

To.  Heights. 

A 

0.00 

B 

2.00 

6.00 

4.00 

-  4.00 

C 

6.00 

7.00 

1.00 

—  5.00 

D 

7.00 

6.00 

1.00 

-  4.00 

E 

9.00 

4.00 

5.00 

+  1.00 

F 

4.00 

8.00 

4.00 

—  3.00 

+  28.00 

-  31.00 

3.00 

THE  PRACTICE. 


35 


In  levelling  for  "sections,"  the  distances  between  the 
points  levelled  must  be  recorded.  They  are  usually  put  down 
after  the  stations  to  which  they  are  measured ;  although  in 
surveying  with  the  compass,  etc.,  they  are  put  down  after  the 
stations  from  which  they  are  measured.  In  the  following 
notes,  which  contain  intermediate  stations,  they  are  put  down 
"before  the  stations  to  which  they  are  measured.  It  should  be 
remembered  that  these  distances  are  measured  between  the 
points  at  which  the  rod  is  held,  and  have  no  reference  to  the 
points  at  which  the  instrument  is  set  up  : 


Distance. 

Stations. 

B.  S.  + 

F.  S.  - 

Rise. 

Fall. 

To.  Heights. 

260 

91.397 

100 

261 

4.576 

3.726 

0.850 

92.247 

100 

262 

5.420 

4.500 

0.920 

93.167 

100 

263 

4.500 

3.170 

1.330 

94.497 

40 

263.40 

4.910 

4.938 

0.028 

94.469 

60 

264 

4.938 

6.386 

1.448 

93.021 

100 

265 

3.380 

4.640 

1.260 

91.761 

100 

266 

4.640 

5.400 

0.760 

91.001 

70 

266.70 

2.760 

3.070 

0.310 

90.691 

30 

267 

3.070 

3.750 

0.680 

90.011 

100 

268 

3.750 

6.925 

3.175 

86.836 

41.944 

46.505 

—  4.561 

41.944 

+91.397 

—  4.561 

86.836 

(55.)  Second  Form  of  Field-book.  This  is  presented  below. 
It  refers  to  the  same  stations  and  levels  noted  in  the  first 
table,  and  shown  in  Fig.  50  : 


Stations. 

Distances. 

Back-sights. 

Ht.  Inst.  above  Datum. 

Fore-sights. 

To.  Heights. 

A 

0.00 

B 

100 

2.00 

+  2.00 

6.00 

—  4.00 

C 

60 

3.00 

—  1.00 

4.00 

-  5.00 

D 

40 

2.00 

—  3.00 

1.00 

-  4.00 

E 

70 

6.00 

+  2.00 

1.00 

+  1.00 

F 

50 

2.00 

+  3.00 

6.00 

-  3.00 

15.00 

18.00 

—  3.00 

36         LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

In  the  preceding  form  it  will  be  seen  that  a  new  column  is 
introduced,  containing  the  Height  of  the  Instrument  (i.  e.,  of 
its  line  of  sight),  not  above  the  ground  where  it  stands,  but 
above  the  Datum,  or  starting-point,  of  the  levels.  The  former 
columns  of  "  Rise  "  and  "  Fall "  are  omitted.  The  preceding 
notes  are  taken  thus:  The  height  of  the  starting-point,  or 
"  datum,"  at  A,  is  0.00.  The  instrument  being  set  up  and 
levelled,  the  rod  is  held  at  A.  The  back-sight  upon  it  is  2.00 ; 
therefore  the  height  of  the  instrument  is  also  2.00.  The  rod 
is  next  held  at  B.  The  fore-sight  to  it  is  6.00.  That  point  is 
therefore  6.00  below  the  instrument,  or  2.00  —  6.00  =  —  4.00 
below  the  datum.  The  instrument  is  now  moved,  and  again 
set  up,  and  the  back-sight  to  B,  being  3.00,  the  height  of  the 
instrument  is  —  4.00  +  3.00  =  —  1.00,  and  so  on ;  the  height 
of  the  instrument  being  always  obtained  by  adding  the  back- 
sight to  the  height  of  the  peg  on  which  the  rod  is  held,  and 
the  height  of  the  next  peg  being  obtained  by  subtracting  the 
fore-sight  to  the  rod  held  on  that  peg,  from  the  height  of  the 
instrument. 

This  form  is  better  than  the  first  form,  in  levelling  for  a 
section  of  the  ground  to  make  a  profile;  or  when  several  ob- 
servations are  to  be  made  at  one  setting  of  the  level ;  or  when 
points  of  desired  heights  are  to  be  established,  as  in  "  Level- 
ling-location,"  Chapter  VIII. 

This  form  may  be  modified  by  putting  the  back-sights  on 
the  same  line  with  the  stations  to  which  they  are  taken.  This 
avoids  the  defect  of  the  first  form,  but  introduces  the  new 
defect  of  writing  them  down  after  the  number  which  they 
precede,  in  a  back-handed  way,  which  maybe  a  source  of 
error. 

This  modification  is  shown  in  the  following  table,  which 
corresponds  to  Fig.  51.  In  the  column  of  fore-sights,  the 
"  turning-points  "  (T.  P.),  and  "  interme'diate  points  "  (Int.), 
are  put  in  separate  columns ;  so  that,  to  prove  the  work,  the 
difference  of  the  sum  of  the  back-sights  and  of  the  sum  of  the 
turning-point  fore-sights,  is  the  number  which  should  equal 
the  difference  of  the  heights  of  the  first  and  last  points : 


AV  9 


// 


THE   PRACTICE. 


37 


F.  S.  - 

Stations. 

Distances. 

B.  S.  + 

Ht.  of  Inst. 

T.  P. 

Int. 

To.  Heights. 

A 

2.00 

0.00 

B 

+  2.00 

6.00 

—  4.00 

C 

7.00 

—  5.00 

D 

9.00 

6.00 

—  4.00 

E 

+  5.00 

4.00 

+  1.00 

F 

8.00 

-  3.00 

+  11.00 

-  14.00 

+  11.00 

-3.00 

When  a  line  is  divided  up  into  stations  of  100  feet  each, 
as  on  railroad  work,  the  number  of  the  station  indicates  its 
distance  from  the  starting-point.  "When  an  observation  is 
taken  at  a  point  between  these  hundred-feet  stations,  it  is  noted 
as  a  decimal  thus:  Station  4.60  is  460  feet  from  the  starting- 
point.  In  the  field-notes  of  such  work,  the  column  of  distances 
may  be  omitted,  as  in  the  following  table.  The  heights  and 
distances  are  the  same  as  in  the  last  table  under  Art.  (54) : 


Stations. 

B.  S.  + 

Ht.  of  Inst. 

P.  S.  — 

Total  Heights. 

T.  P. 

Int. 

260 

4.576 

95.973 

91.397 

261 

5.420 

97.667 

3.726 

92^247 

262 

4.500 

93.167 

263 

4?910 

99.407 

3.170 

94.497 

263.40 

4.938 

94.469 

264 

3.380 

96.401 

6.386 

93.021 

265 

4.640 

91.761 

266 

2.760 

93.761 

5.400 

• 

91.001 

266.70 

3.070 

90.691 

267 

3.750 

90.011 

268 

6.925 

86.836 

+  21.046 

-  25.607 

+  21.046 

_ 

—  4.561 

+  91:397. 

+  86.836 

G 


7 


38 


LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


(56.)  Third  Form  of  Field-book.  In  this  form,  the  defects 
of  the  preceding  methods  are  avoided,'  and  it  approximates  to 
a  sketch  of  the  operations ;  the  fore-sights  being  placed  before 
the  stations  to  which  they  are  taken,  and  the  back-sights  after 
them.  The  distances  are  placed  before  the  stations  to  which 
they  are  taken ;  or  after  those  from  which  they  are  taken. 
Another  advantage  is  that  the  stations,  their  heights,  and  the 
distances,  are  brought  together ;  which  facilitates  the  making 
of  a  profile.  The  following  table  is  the  case  given  in  Fig.  50 : 


F.  S.  -  " 

JDistances. 

Stations. 

Ht.  of  Peg. 

B.  S.  + 

Ht.  of  Inst. 

A 

o.oo- 

2.00 

+  2.00 

6.00 

100 

B 

—  4.00 

3.00 

—  1.00 

4.00 

60 

C 

—  5.00 

2.00 

—  3.00 

1.00 

40 

D 

—  4.00 

6.00 

+  2.00 

1.00 

TO 

E 

4-  1.00 

2.00 

'  +  3.00 

6.00 

50 

F 

—  3.00 

-  18.00 

• 

+  15.00 
—  18.00 

—    3.00 

When  "  intermediates  "  are  taken,  the  first  column  may  be 
divided  into  two  heads  (as  in  the  second  table,  Art.  55),  re- 
spectively "turning-points"  (T.  P.),  and  "  intermediate  points" 
(Int.).  The  work  is  tested  by  taking  the  difference  of  the  sum 
of  the  "  T.  P.'s  "  and  "  B.  S.'s  "  The  symbol  0  is  used  to  rep- 
resent the  height  of  the  cross-hairs.  This  table  is  for  Fig.  51 : 


F.  S.  - 

T.  P. 

Int. 

Stations. 

Distances. 

Ht.  of  Peg. 

B.  S.  + 

© 

A 

100 

0.00 

2.00 

+  2.00 

6.00* 

B 

60 

—  4.00 

7.00 

C 

40 

—  5.00 

6.00 

D 

70 

-4.00 

9.00 

+  5.00 

4.00 

E 

50 

+  1.00 

8.00 

F 

-3.00 

-  14.00 

+  11.00- 

—  14.00 

—    3.00 

THE  PRACTICE. 

Fourth  Form  of  Field-book.  In  this  the  back 
placed  directly  under  the  height  of  the  station  to  which 
are  taken,  which  lessens  the  chance  of  making  mistakes  in 
adding  to  get  the  height  of  instrument.  The  height  of  in- 
strument is  distinguished  by  being  included  between  two 
horizontal  lines.  The  following  table  refers  to  Fig.  51. 


Station. 

F.  S. 

Heights. 

Remarks. 

A 

B 
C 
D 

E 
F 

6.00 
7.00 
6.00 

4.00 
8.00 

0.00 

2.00 

2.00 

-4.00 
—  5.00 
-4.00 
9.00 

5.00 

1.00 
—3.00 

(57.)  Best  Length  of  Sights.  There  are  two  classes  of  inac- 
curacies. With  very  long  sights,  the  errors  of  imperfect  ad- 
justment and  curvature  are  greatest ;  the  former  varying  as 
the  length,  and  the  latter  as  the  square  of  the  length.  With 
very  short  sights,  and  therefore  more  numerous,  the  errors  of 
inaccurate  sighting  at  the  target  are  greatest.  The  best  usual 
mean  is  from  200  feet  to  300  feet,  or  more  if  equal  distances 
for  back-sights  and  fore-sights  to  turning-points  can  be  ob- 
tained. 

(58.)  Equal  Distances  of  Sight.  They  are  always  very  de- 
sirable. They  are  most  easily  determined,  when  no  stakes 
have  been  previously  set,  by  "  stadia  "  cross-hairs  in  the  tele- 
scope of  the  level.  [L.  S.,  375.] 

(59.)  Datum-Level  This  is  the  plane  of  reference,  from 
.which,  above  it  or  below  it,  usually  the  former,  the  heights  of 
all  points  of  the  line  are-  reckoned. 


40         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

It  may  be  taken  as  the  height  of  the  starting-point.     If 
the  line  descends,  it  is  better  to  call  the  starting-point  10  feet 
or  100  feet  above  some  imaginary  plane,  so  that  points  below 
*  the  starting-point  may  not  have  minus  signs. 

It  is  desirable  to  refer  all  levels  in  a  country  to  some  one 
datum.  This  is  usually  the  surface  of  the  sea,  and  for  general 
purposes  mean  tide  is  best.  Low-water  mark  should  be  the 
datum  when  the  le  veilings  are  connected  with  harbor-surveys, 
whose  soundings  always  refer  to  low  water.  High-water 
mark  should  be  used  when  the  levellings  relate  to  the' drain 
age  of  a  country. 

(60.)  Bench-Marks  (B,  M.X     These  are  permanent  objects, 
natural  or  artificial,  whose  heights  above  the  datum  are  de- 
.  termined  and  recorded  for  future  reference. 

Good  objects  are  these :  pointed  tops  of  rocks ;  tops  of 
milestones;  stone  door-sills;  tops  of  gate-posts  or  hinges;  and 
generally  any  object  not  easily  disturbed,  and  easily  described 
and  found. 

A  knob  made  on  the  spreading  root  of  a  tree  is  good.  A 
nail  may  be  driven  in  it,  and  the  FlG  52 

tree  "blazed"  and    marked,  as   in 
Fig.  52.     A  stake  will  do  till  frost. 

Bench-marks  should  be  made 
near  the  starting-  point  of  a  line  of 
levels ;  near  where  the  line  crosses  a 
road  ;  on  each  side  of  a  river  crossed 
by  it ;  at  the  top  and  bottom  of  any 
high  hill  passed  over ;  and  always  at  every  half-mile  or  mile. 

The  precise  location  and  description  of  every  B.  M.  should 
be  noted  very  fully  and  precisely,  and  in  such  a  way  that  an 
entire  stranger  could  find  it,  with  the  aid  of  the  notes. 

v\> 

(81.)  Check-Levels,  or  Test-Levels.  No  single  set  of  levels 
is  to  be  trusted ;  but  they  must  be  tested  by  another  set,  run 
between  the  bench-marks  (B.  M.'s),  though  not  necessarily 
over  the  same  ground. 


THE  PRACTICE.  4-! 

A  set  of  levels  will  verify  themselves  if  they  come  around 
to  the  starting-point  again. 

(62.)  Limits  of  Precision.  Errors  and  inaccuracies  should 
be  carefully  distinguished.  For  the  latter,  every  leveller 
must  make  a  standard  for  himself,  so  as  to  be  able,  in  testing 
his  work,  to  distinguish  any  real  error  from  his  usual  inac- 
curacy. 

The  result  of  four  sets  of  levellings,  in  France,  of  from  45 
to  140  miles,  averaged  a  difference  of  -fa  ft.  in  43  miles,  and 
the  greatest  error  was  £  ft.  in  56  miles. 

A  French  leveller,  M.  Bourdaloue,  contracts  to  level  the 
B.  M.'s  of  a  K.  R.  survey  to  within  0.002  ft.  per  mile,  or  -fa  ft. 
per  50  miles. 

In  Scotland,  the  difference  of  two  sets  of  levels  of  26  miles 
was  0.02  ft. 

(63.)  Trial-Levels,  or  Flying-Levels.  Their  object  is  to  get 
a  general  approximate  idea  of  the  comparative  heights  of 
a  portion  of  the  country,  as  a  guide  in  choosing  lines  to  be 
levelled  more  accurately.  More  rapidity  is  required,  and  less 
precision  is  necessary.  The  distances  may  be  measured  at  the 
same  time  by  stadia-hairs. 

(64.)  Levelling  for  Sections.  The  object  of  this  is  to  meas- 
ure all  the  ascents  and  descents  of  the  line,  and  the  distances 
between  the  points  at  which  the  slope  changes;  so  that  a 
section  or  profile  of  it  can  be  made  from  the  observations  taken. 

The  line  of  a  railroad  is  usually  set  out  by  a  party  with 
compass  or  transit,  who  drive  at  every  hundred  feet  a  large 
stake  with  the  number  of  the  station  on  it,  and  beside  it  a 
small  level-peg,  even  with  the  surface  of  the  ground.  On  this 
the  rod  is  held  for  the  observations.  The  level-peg  is  set  in 
"  line,"  and  the  large  stake  a  foot  or  two  to  one  side. 

(65.)  Profiles.  A  profile  is  a  section  of  ground  by  a  verti- 
cal plane  or  cylindrical  surface/  passing  through  the  line  along 

1  A  cylindrical  surface  is  here  understood  to  mean  that  formed  by  a  line  mov- 
ing parallel  to  itself  along  any  line,  instead  of  only  a  circle,  as  in  elementary 
geometry. 


4:2          LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

which  a  profile  is  desired.  It  represents  to  any  desired  scale 
the  heights  and  distances  of  the  various  points  of  a  line,  its 
ascents  and  descents,  as  seen  in  a  side  view.  It  is  made  thus : 
Any  point  on  the  paper  being  assumed  for  the  first  station,  a 
horizontal  line  is  drawn  through  it ;  the  distance  to  the  next 
station  is  measured  along  it,  to  the  required  scale ;  at  the  ter- 
mination of  this  distance  a  vertical  line  is  drawn ;  and  the 
given  height  of  the  second  station  above  or  below  the  first  is 
set  off  on  this  vertical  line.  The  point  thus  fixed  determines 
the  second  station,  and  a  line  joining  it  to  the  first  station 
represents  the  slope  of  the  ground  between  the  two.  The  pro- 
cess is  repeated  for  the  next  station,  etc. 

But  the  rises  and  falls  of  a  line  are  always  very  small  in 
proportion  to  the  distances  passed  over,  even  mountains  being, 
merely  as  the  roughnesses  of  the  rind  of  an  orange.  If  the  dis- 
tances and  the  heights  were  represented  on  a  profile  to  the 
same  scale,  the  latter  would  be  hardly  visible.  To  make  them 
more  apparent,  it  is  usual  to  "  exaggerate  the  vertical  scale  " 
tenfold,  or  more ;  i.  e.,  to  make  the  representation  of  a  foot 
of  height  ten  times  as  great  as  that  of  a  foot  of  length,  as  in 
Fig.  50,  in  .which  one  inch  represents  one  hundred  feet  for 
the  distances,  and  ten  feet  for  the  heights. 

In  practice,  engraved  profile-paper  is  generally  used,  which 
is  ruled  in  squares  or  rectangles,  to  which  any  arbitrary  values 
may  be  assigned. 

When  the  line  levelled  over  is  not  straight,  the  profile, 
whose  length  is  that  of  the  line  straightened  out,  will  extend 
beyond  the  "  plan  "  when  both  are  on  the  same  sheet. 

(66.)  Cross-Levels.  These  show  the  heights  of  the  ground 
on  a  line  at  right  angles  to  the  main  line.  They  give  "  cross- 
sections  "  of  it.  In  the  note-book  they  are  put  on  the  right- 
hand  page.  They  may  be  taken  at  the  same  time  with  the 
other  levels,  or  independently.  In  taking  cross-levels  where 
the  slopes  are  quite  steep,  as  in  mountain  districts,  frequent 
settings  of  the  instrument  are  necess'ary. 

A  much  more  rapid  method  is  by  the  use  of  "  cross-sec- 


DIFFICULTIES. 


tion  rods."  These  are  two  rods,  one  of  which  is  about  ten  or 
twelve  feet  long,  provided  with  a  bubble-tube  near  each  end, 
so  as  to  be  held  level,  and  graduated  to  feet,  tenths,  and  hun- 
dredths.  The  other  is  simply  a  graduated  rod.  The  manner 
of  using  them  is  shown  in  Fig.  53. 


FIG.  53. 


Aslope-level  is  sometimes  used.     See  "Angular  Survey- 
ing," Part  II. 


CHAPTER  VII. 

DIFFICULTIES. 

i 

(67.)  Steep  Slopes,  In  descending  or  ascending  a  hill,  the 
instrument  and  the  rod  should  be  so  placed  that  the  sight 
should  strike  as  near  as  possible  to  the  bottom  of  the  rod  on 
the  up-hill  side,  and  the  top  of  the  rod  on  the  down-hill  side. 

Try  this  by  levelling  over  two  screws,  setting  the  instru- 
ment so  that  one  pair  of  opposite  plate-screws  shall  point  m 
the  direction  of  the  line,  but  do  not  be  too  particular ;  it  is  a 
waste  of  time. 

Doing  this  produces  sights  of  unequal  length.  The  rod 
being  about  twice  as  high  as  the  instrument,  the  down-hill 
sights  will  be  about  double  the  length  of  the  up-hill  ones,  as 
shown  in  Fiff.  54.  Then  set  to  one  side  of  the  line.  This  is 


-J4         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

FIG.  54. 


necessary  on  slopes  so  steep  that  the  rod  is  too  near  the  level 
to  be  read.  If  this  be  impossible,  keep  notes  of  the  lengths 
of  the  sights  to  the  turning-points,  backward  and  forward, 
and  as  soon  as  possible  take  sights  unequal  in  the  contrary 
direction  till  the  differences  ot  lengths  balance  the  former 
ones.  When  approaching  a  long  ascent  or  descent,  make 
these  compensations  in  advance. 

In  levelling  over  a  line  of  stakes  already  set,  as  on  a  rail- 
road, at  every  100  ft.,  if  the  line  of  sight  strikes  not  quite  up 
to  one,  drive  a  peg  as  high  as  you  can  see  it,  and  make  it  a 
turning-point,  noting  it  "  peg  "  in  the  field-book. 

In  levelling  across  a  hill  or  hollow,  instead  of  setting  the 
instrument  on  the  top  of  the  hill  or  bottom  of  the  hollow,  time 
will  be  saved  by  the  method  represented  in  Figs.  55  and  56. 


FIG.  55. 


(68.)  When  the  rod  is  a  little  too  low,  raise  it  alongside  of 
a  stake,  or  the  body,  and  put  the  top  of  the  rod  "right;" 
then  measure  down  from  the  bottom  of  the  rod,  and  add  it  to 

Q 

its  length. 


DIFFICULTIES. 
FIG.  58. 


(69.)  When  the  rod  is  a  little  too  high,  so  that  the  line  of 
sight  strikes  the  peg  below  the  bottom  of  the  rod,  measure 
down  from  the  top  of  the  peg,  and  put  down  the  sight  with  a 
contrary  sign  to  what  it  would  have  had ;  i.  e.,  if  a  back-sight 
make  it  minus,  and  if  a  fore-sight  make  it  plus. 

(70.)  When  the  rod  is  too  near.  When  no  figure  is  visi- 
ble, raise  the  rod  slowly  till  a  figure  comes  in  sight.  If  too 
near  to  read,  and  there  is  no  target,  use  a  field-book  as  target. 
If  the  instrument  is  exactly  over  the  peg,  measure  up  to  the 
height  of  the  cross- hairs,  as  given  by  the  side-screws. 

(71.)  WATEK.  A. — A  pond  too  wide  to  le  sighted  across. 
Drive  a  peg  to  the  level  of  the  water,  on  the  first  side,  and 
observe  its  height,  as  an  F.  S.  Then  drive  a  peg  on  the  other 
side  of  the  pond,  also  to  the  surface  of  the  water.  Hold  the 
rod  on  it.  Set  up  the  level  beyond  it,  and  sight  to  it  as  a 
B.  S.,  and  put  down  the  observation  as  if  it  had  been  taken 
to  the  first  peg. 

FIG.  57. 


46        LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


F.  S. 

Sta. 

Ht. 

B.  S. 

e 

74 

60.00 

3.00 

53.00 

5.0 

74.89  ) 
81.89  f 

48.00 

% 

6.00 

54.00 

There  must  be  no  wind  in  the  direction  of  the  line  of  level. 

J3. — For  levelling  'across  a  running  stream.  Set  the  two 
pegs  in  a  line  at  right  angles  to  the  current,  although  the  line 
to  be  levelled  may  cross  it  obliquely. 

If  a  profile  or  section  of  the  ground  under  the  water  be  re- 
quired, find  the  height  of  the  surface,  and  measure  the  depths 
below  this  at  a  sufficient  number  of  points,  measuring  the 
distances  also,  and  put  these  depths  down  as  fore-sights. 

(72.)  A  Swamp,  or  Marsh.  This  cannot  be  treated  like  a 
pond,  for  the  water  may  seem  nearly  stagnant  while  its  sur- 
face has  considerable  slope,  its  flow  being  retarded  by  vegeta- 
tion. If  only  slightly  "  shaky,"  have  an  observer  at  each  end 
of  the  level.  If  more  so,  push  the  legs  down  as  far  as  they 
will  go,  and  let  both  observers  lie  down  on  their  sides.  If  still 
more  "  shaky,"  drive  three  stakes  or  piles,  to  support  the  legs 
of  the  tripod,  and  stand  the  tripod  on  them. 

A  water-level  will  level  itself.  Use  that  for  intermediate 
points  on  the  swamp,  and  test  the  result  by  levelling  around 
the  swamp  with  the  spirit-level. 

(73.)  Underwood.  If  it  cannot  be  cut  away,  set  the  instru- 
ment on  some  eminence,  natural  or  artificial. 

(74.)  Board  Fence.  Eun  a  knife-blade  through  one  of  the 
boards,  and  hold  the  rod  upon  it  on  each  side  of  the  fence,  as 
if  it  were  a  peg,  keeping  the  blade  in  the  same  horizontal 
position  while  the  rod  and  instrument  are  taken  over. 

(75.)  A  Wall,  First  Method.  Drive  a  peg  at  the  bottom  of 
the  wall,  on  the  first  side,  and  observe  on  it.  Measure  the 
height  of  the  wall  above  the  peg,  and  put  this  down  as  a  B.  S. 
Drive  another  peg  on  the  other  side  of  the  wall ;  measure  down 


DIFFICULTIES.  47 

to  it  from  the  top  of  the  wall,  and  put  that  down  as  an  F.  S., 
just  as  if  the  level  had  been  set  in  the  air  at  the  height  of  the 
top  of  the  wall,  and  this  B.  S.  and  F.  S.  had  been  really  taken. 
Set  up  the  instrument  beyond  the  wall,  take  a  B.  S.  to  this 
peg,  and  go  on  as  usual. 

FIG.  58. 


F.  S. 

Sta. 

Ht. 

B  S. 

e 

50 

74.00 

5.00 

79.00 

3.00 

Peg. 

76.00 

13.00 

89.00 

12.00 

Pe|. 

77.00 

2.00 

79.00 

1.00 

51 

78.00 

Second  Method.     Mark  where  the  line  of  sight  strikes  the 
wall ;  measure  up  to  the  top  of  the  wall,  and  put  this  down  as 
an  F.  S.,  with  a  plus  sign,  as  in  (69),  where  the  line  of  sight, 
struck  below  the  top  of  the  peg. 

On  the  other  side  of  the  wall,  sight  back  to  it,  and  mark 
where  the  line  of  sight  strikes.  Measure  to  the  top  of  the 
wall,  and  put  this  down  as  a  B.  S.,  with  a  minus  sign,  and 
then  go  on  as  usual. 

(76.)  House.  First  try  to  find  some  place  for  the  instru- 
ment from  which  you  can  see  through,  by  opening  doors  or 
windows.  Or,  find  some  place  in  the  house  where  you  can 
set  the  instrument  and  see  both  ways,  or  hold  the  rod  at  some 
point  *nside,  and  look  to  it  from  front  and  back.  A  straight 
stick  may  be  used  if  the  rod  cannot  be  held  upright,  and  the 
height  measured  on  the  rod.. 

(77.)  The  Sun.     It  often  causes  the  leveller  much  difficulty. 
1.  By  shining  in  the  object-glass.     If  the  instrument  has  a 


48        LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

shade  on  it,  draw  it  out.     If  not,  shade  the  glass  with  your 
hand  or  hat,  or  set  the  instrument  to  one  side  of  the  line. 

2.  By  heating  the  level  unequally  in  all  its  parts.     Hold- 
ing an  umbrella  over  it  will  remedy  this. 

3.  By  causing  irregular  refraction.      Some  parts  of  the 
ground  become  heated  more  than  others,  and  therefore  rarefy 
the  air  at  those  places.     This  cannot  be  avoided  nor  corrected. 

(78.)  Wind.  "Watch  for  lulls  of  wind,  and  observe  then  sev- 
eral times,  and  take  the  mean.  The  least  wind  is  at  daybreak. 

(79.)  Idiosyncrasies;  Different  persons  do  not  see  things 
precisely  alike.  Each  individual  may  have  an  inaccuracy 
peculiar  to  himself.  One  may  read  an  observation  higher  or 
lower  than  another  equal  in  skill.  Also,  a  person's  right  and 
left  eye  may  differ.  This  difference  in  individuals  is  termed 
their  "  personal  equation." 

To  test  the  accuracy  of  your  eye,  turn  the  head  so  as  to 
bring  the  eyes  in  the  same  vertical  line,  and  sight  to  the  rod 
held  horizontally.  Note  where  the  vertical  hair  strikes.  Then 
turn  the  head  to  the  other  side,  so  as  to  invert  the  position  of 
the  eyes,  and  then  sight  again.  As  before,  the  mean  of  the 
two  readings  is  the  correct  one. 

(80.)  Reciprocal  Levelling,  This  is  to  be  used  when  it  is 
impossible  to  set  midway  between  the  two  points. 

FIG.  59. 


Set  the  instrument  over  A,  and  sight  to  rod  at  B,  and  note 
reading.     The  difference  of  the  reading  and  of  the  height  of 


LEVELLING   LOCATION.  49 

the  cross-hairs  gives  a  difference  of  height  of  A  and  B.  Tnen  set 
up  at  B,  and  observe  to  A,  similarly.  A  new  difference  of 
height  is  obtained.  The  mean  of  these  two  is  the  correct  one. 

Ht.  of  cross-hairs  above  peg  at  A  =  4'.3     Ht.  of  cross-hairs  above  peg  at  B  =  4'.9 
Observation  to  B  =  7'.0  Observation  to  A  =  4'. 2 

Diff.  of  height     =  2'.7  Diff.  of  height     =  O'.Y 

True  difference  -  i  (2'.7  +  0'.7)  =  I'.V. 

Otherwise,  set  the  instrument  at  an  equal  distance  from 
each  point,  as  A'  and  B',  and  observe  to  each  in  turn.  The 
mean  of  the  two  differences  of  height  obtained  will  be  the  true 
difference.,  as  before. 


CHAPTER  VIII. 

LEVELLING     LOCATION. 

(81.)  Its  Nature.  It  is  the  converse  of  the  general  problem 
of  levelling,  which  is  to  find  the  difference  of  heights  of  two 
given  points.  This  consists  in  determining  the  place  of  a 
point  of  any  required  height  above  or  below  any  given  point. 

To  do  this,  hold  the  rod  on  some  point  of  known  height 
above  the  datum-level ;  sight  to  it,  and  thus  determine  the 
height  of  the  cross-hairs.  Subtract  from  this  the  desired 
height  of  the  required  point,  and  set  the  target  at  the  differ- 
ence. Hold  the  rod  at  the  place  where  the  height  is  desired, 
and  raise  or  lower  it  till  the  cross-hair  bisects  the  target.  Then 
the  bottom  of  the  rod  is  at  the  desired  height.  Usually,  a  peg 
is  driven  till  its  top  is  at  the  given  height  above  the  datum. 

(82.)  Difficulties.  If  the  difference  of  height  be  too  much 
to  be  measured  at  one  setting  of  the  instrument,  take  a  series 


50         LEVELLING,  TOPOGRAPHY,  AND   HIGHER   SURVEYING. 


FIG.  60. 


of  levels  up  or  down  to  the  desired  point.     So,  too,  if  they  be 
far  apart ;  and  thus  find  a  place  where,  the  instrument  having 
.  a  known  height  of  cross-hairs,  the  target  can  finally  be  set,  as 
before. 

If  the  ground  be  so  low  or  so  high  that  a  peg  cannot  be 
set  with  its  top  at  the  required  height,  drive  a  peg  till  its  top 
is  just  above  the  surface  of  the  ground.  Observe  to  the  rod 
on  it,  determine  its  height  above  or  below  the  desired  point, 
and  note  this  on  a  large  stake  driven  beside  it ;  or,  place  its 
top  a  whole  number  of  feet  above  or  below  the  required  height, 
and  mark  the  difference  on  it,  or  on  a  stake  beside  it. 

(83.)  Staking  out  Work,  "When  embankments  and  excava- 
tions are  to  be  made  for  roads,  etc.,  side-stakes  are  set  at  points 
in  their  intended  outside 
edges;  i.  e.,  where  their 
slopes  will  meet  the  sur- 
face of  the  ground;  and 
the  height  which  the 
ground  at  those  points  is 
above  or  below  the  re- 
quired height  or  depth  of 
the  top  or  bottom  of  the 
finished  work,  is  marked  on  these  stakes  with  the  words 
"  cut,"  or  "  fill,"  or  the  signs  -f  or  — . 

The  places  of  the  stakes 
are  found  by  trial.  (See 
Gillespie's  Koad-making, 
p.  145.)  These  stakes  are 
set  to  prepare  the  work 
for  contractors.  "When  the 
work  is  nearly  finished, 
other  stakes  are  set  at  the 
exact  required  height. 

In  staking  out  foundation-pits ',  set  temporary  stakes  ex- 
actly above  the  intended  bottom  angles  of  the  completed  pit, 
thus  marking  out  on  the  surface  of  the  ground  its  intended 


fill  12  or"- 


FIG.  61. 


5  or  *  5 


LEVELLING   LOCATION.  51 

shape.  Take  the  heights  of  each  of  these  stakes  and  move 
them  outward  such  distances  that  cutting  down  from  them 
with  the  proper  depth  and  slope  will  bring  you  to  the  desired 
bottom  angle. 

(84.)  To  locate  a  Level-Line,  This  consists  in  determining 
on  the  surface  of  the  ground  a  series  of  points  which  are  at 
the  same  level;  i.  e.,  at  the  same  height  above  some  datum. 
Set  one  peg  at  the  desired  height,  as  in  (81).  Sight  to  the  rod 
held  thereon,  and  make  fast  the  target  when  bisected.  Then 
send  on  the  rod  in  the  desired  direction,  and  have  it  moved  up 
or  down  along  the  slope  of  the  ground,  until  the  target  is 
again  bisected.  This  gives  a  second  point.  So  go  on  as  far 
as  sights  can  be  correctly  taken,  keeping  unchanged  the  in- 
strument and  target.  Make  the  last  point  sighted  to  a  "  turn- 
ing-point." Carry  the  instrument  beyond  it,  set  up  again, 
take  a  B.  S.,  and  proceed  as  at  first. 

The  rod  should  be  held  and  pegs  driven  at  points  so  near 
together  that  the  level-line  between  them  will  be  approxi- 
mately straight. 

(85.)  Applications.  One  use  of  this  operation  is  to  mark 
out  the  line  which  will  be  the  edge  of  the  water  of  a  pond  to 
be  formed  by  a  dam.  In  that  case,  a  point  of  a  height  equal 
to  that  of  the  top  of  the  proposed  dam,  plus  the  height  which 
the  water  will  stand  on  it  (to  be  determined  by  hydraulic 
formulas),  will  be  the  starting-point.  Then  proceed  to  set 
stakes  as  directed  in  the  last  article. 

The  line  from  stake  to  stake  may  then  be  surveyed  like 
the  sides  of  a  field,  and  the  area  to  be  overflowed  thus  de- 
termined. 

Strictly,  the  surface  of  the  water  behind  a  dam  is  not  level, 
but  is  curved  concavely  upward,  and  is  therefore  higher  and 
sets  back  farther  than  if  level.  This  backing  up  of  the  water 
is  called  Remous. 

Another  important  application  of  this  problem  is  to  obtain 
"  contour  lines  "  for  Topography. 


52         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

(86.)  To  run  a  Grade-Line.  This  consists  in  setting  a  series 
of  pegs  so  that  their  tops  shall  be  points  in  a  line  which  shall 
have  any  required  slope,  ascending  or  descending. 

When  a  grade-line  is  to  be  run  straight  between  two  given 
points,  set  the  level  over  one  point,-  set  the  target  at  the 
height  of  the  cross-hairs,  hold  the  rod  on  the  other  point,  and 
raise  or  lower  one  end  of  the  instrument  till  the  cross-hair 
bisects  the  target.  Then  send  the  rod  along  the  line,  and 
drive  pegs  to  such  heights  that  when  the  rod  is  held  on  them 
the  cross-hair  will  bisect  the  target.  A  stake  may  be  driven 
at  the  extreme  point  to  the  height  of  the  target. 

A  line  of  uniform 

,  ,  .  FIG.  62. 

grade  or  slope  is  not  a 
straight  line.  Calling 
the  globe  spherical, 
this  line,  when  traced 
in  the  plane  of  a  great 
circle,  would  be  a  log- 
arithmic spiral.  On  a 
length  of  six  miles,  the  difference  in  the  middle  between  it  and 
its  straight  chord  would  be  six  feet. 


PART  II. 
INDIRECT    LEVELLING. 


CHAPTEE  I. 

METHODS     AND     INSTRUMENTS. 

(87.)  Vertical  Surveying.  Levelling  may  be  named  VER-. 
TICAL  SURVEYING,  or  Up-and-down  Surveying ;  Land  Sur- 
veying being  HORIZONTAL  SURVEYING,  or  Right-and-left  and 
Fore-and-aft  Surveying. 

AH  the  methods  of  determining  the  position  of  a. point  in 
horizontal  surveying,  may  be  used  in  vertical  surveying. 

The  point  may  be  determined  by  coordinates  situated  in 
a  vertical  plane,  as  in  any  of  the  systems  employed  in  L.  S. 
(Part  I.,  Chapter  I.),  in  a  horizontal  plane. 

Pio  M  Thus,  if  a  balloon  be  held  down  by  a  sin- 

gle rope  attached  to  a  point  in  a  level  sur- 
face, its  height  above  that  surface  is  found 
by  measuring  the  length  of  the  rope.  This 
is  the  Direct  Method.  It  resembles  that 
of  "  rectangular  coordinates,"  L.  S.  (6) ; 
though  here  only  one  of  the  coordinates  is  measured.  The 
other  might  be  situated  anywhere  in  the  surface. 

If,  however,  the  balloon  be  held  down 
by  two  cords,  its  height  can  be  determined 
by  measuring  the  length  of  the  cords  and 
the  distance  between  their  lower  ends.  They 
correspond  to  the  "focal  coordinates"  of 
L.  S.  (5).  The  required  vertical  height  can 


LEVELLING,  TOPOGRAPHY,  AXD   HIGHER  SURVEYING. 


FIG.  65. 


Fie. 


"  angular 


CO- 


FIG.  67. 


be  calculated  by  trigonometry.      So  in  the  following  other 
Indirect  Methods. 

The  length  of  the  string  of  a  kite,  and 
the  angle  which  this  string  makes  with 
the  horizon,  are  the  "polar  coordinates  " 
of  the  kite ;  as  in  L.  S.  (7). 

The  "  angles  of  elevation  "  of  a  me- 
teor, observed  by  two  persons  in  the 
same  vertical  plane  with  it,  and  at  known 
distances   apart,   are   its 
ordinates,"  as  in  L.  S.  (8). 

Finally,  an  aeronaut  could  determine 
his  own  height  by  observing  the  angles 
subtended  by  three  given  objects  situ- 
.ated  on  the  earth's  surface,  at  known 
distances,  and  in  the  same  vertical  plane 
with  him.  These  angles  would  be  the 
"  trilinear  coordinates  "  of  L.  S.  (10). 

Many  other  systems  of  coordinate  lines  and  angles,  va- 
riously combined,  may  be  employed. 

The  desired  heights  may  also  be  determined  by  various 
other  methods,  analogous  to.  those  given  in  L.  S.  for  "  inac- 
cessible distances." 

Combinations  of  measurements  not  in  the  same  vertical 
plane  may  also  be  used,  as  will  be  shown  in  Chapter  III. 

(88.)  Veitical  Angles.    The  vertical  angles  measured  may 
be  those  made — either  with  a  level 
line,  or   with   a  vertical    line — by 
the  line  passing  from  one  point  to 
the  other. 

The  angle  BAC  is  called  an 
"  angle  of  elevation,"  and  the  angle 
B'AC  an  "angle  of  depression." 
The  former  angle  may  be  called  pos- 
itive, and  the  latter  negative. 

The  angle  BAZ  or  B'AZ  is 


/ 


METHODS  AND  INSTRUMENTS. 


55 


called  the  zenith  distance  of  the  object.  It  is  the  complement 
of  the  former  angle,  i.  e.,  =  90°—  that  angle  taken  with  its 
proper  algebraic  sign.  An  angle  of  elevation,  B  A  C  =  10°, 
would  be  a  zenith  distance  of  80°.  An  angle  of  depression, 
B'A  C  =  —  10°,  would  be  a  zenith  distance  of  100°.  The 
zenith  distance  is  preferable  in  important  and  complicated 
operations,  as  avoiding  the  ambiguity  of  the  other  mode  of 
notation. 

(89.)  Instruments.  All  contain  a  divided  circle,  or  arc, 
placed  vertically,  and  a  level  or  plumb  line.  By  these  is 
measured  the  desired  vertical  angle  made  by  the  inclined  line 
with  either  a  level  or  vertical  line. 

This  inclined  line  may  be  an  actual  line  or  a  visual  line. 
In  the  former  case,  it  may  be  a  rod,  or  cord,  or  wire,  as  shown 
in  the  figures : 

FIG.  69. 


FIG.  70. 


FIG.  71. 


This  last  arrangement  of  a  cord  or  wire,  Fig.  71,  is  used 
in  mine  surveying.  A  light  surveyor's  chain  may  be  similarly 
used,  with  the  advantage  of  giving,  at  the  same  time,  differ- 
ence of  heights  and  distance. 


56         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

Diff.  of  hts.  =  length  of  chain  x  sin.  angle. 
Hor.  distance  =  length  of  chain  x  cos.  angle. 

These  instruments  are  all  "  Slope-measurers."  They  are 
also  called  Clinometers,  Clisimeters,  Eclimeters,  etc.,  all  mean- 
ing the  same  thing. 

(90.)  Slopes,  These  may  be  designated  by  their  angles 
with  the  horizon,  or  by  the  relations  of  their  bases  and  heights. 
The  French  engineers  name  a  slope  by  Fw  72 

the  ratio  of  its  height  to  its  base ;  i.  e.,  ^ 

T)  r^ 

-r-~ ;  which  is  the  tangent  of  the  angle 
A  O 

BAG.    The  English  and  Americans  use 
the  ratio  of  the  base  to  the  height ;  i.  e., 

-g-p  >  and  make  the  height  the  unit,  so  that  if  A  C  =  2  C  B, 
the  slope  is  called  2  to  1 ;  and  so  on. 

(91.)  When  the  inclined  line  is  a  visual  line,  such  as  the 
line  of  sight  of  a  telescope,  whose  angular  movements  are 
measured  on  a  vertical  circle  beside  it,  and  when  with  these 
is  combined  a  horizontal  circle  for  measuring  horizontal  an- 
gles, the  instrument  is  called  a  "  Theodolite." 

In  the  usual  American  form,  the  telescope  turns  over.  It 
is  a  transit-theodolite.  (See  Fig.  73.)  It  is  usually  called  sim- 
ply a  "  Transit." 

For  the  usual  English  form,  see  L.  S.,  page  213. 

In  the  usual  French  form,  the  telescope  is  eccentric ;  i.  e., 
on  one  side  of  the  vertical  axis,  and  has  a  counterpoise  on  the 
other  side,  as  in  Fig.  141,  of  mining  transit. 

(92.)  The  Surveyor's  Transit,  Fig.  73.  The  telescope  re- 
volves on  a  horizontal  axis,  which  itself  rests  on  two  stand- 
ards, S  S,  attached  to  the  horizontal  vernier-plate,  H.  The 
graduated  vertical  circle,  A,  by  which  vertical  angles  are 
measured,  is  attached  to  the  telescope  axis,  and  is  read  with  a 
vernier  on  the  lower  side.  A  level,  L,  is  attached  to  the  tele- 
scope, in  the  same  manner  as  that  of  the  Y  level.  The  ver- 


METHODS  AND  INSTRUMENTS. 


57 


nier-plate,  which  carries  the  telesfcope,  is  furnished  with  two 
verniers  on  opposite  sides  of  the  instrument,  and  at  right 
angles  to  the  telescope.  The  vertical  and  the  horizontal 
graduated  circles  are  both  furnished  with  a  clamp  and  slow- 


FIQ.  73. 


motion  screw.  Attached  to  the  upper  parallel  plate  is  another 
clamp,  C,  and  a  pair  of  slow-motion  screws,  T  Ty  by  which 
all  of  the  instrument  above  the  clamp  may  be  given  a  slow 
motion,  horizontally.  The  vernier-plate  is  furnished  with  two 
levels,  at  right  angles  to  each  other.  One  of  them,  D,  is 


58         LEVELLING,  TOPOGRAPHY,  AXD  HIGHER  SURVEYING. 

attached  to  tlie  plate,  and  the  other,  E,  is  fastened  to  the 
standard,  up  out  of  the  way  of  the  second  vernier. 

The  compass  may  be  used  like  a  common  surveyor's  com- 
pass, the  telescope  taking  the  place  of  the  sights.  Its  prin- 
cipal use  is  to  serve  as  a  check  on  the  observations,  the 
difference  of  the  magnetic  bearings  of  two  lines  being  approx- 
imately equal  to  the  angle  measured  between  them  by  the 
more  perfect  instrument. 

The  arrangement  of  the  parts  of  the  telescope,  and  the 
parallel  plates,  are  the  same  as  for  the  Y  level. 

(93.)  Adjustments.  First  Adjustment.  To  cause  the  bub- 
bles to  remain  in  the  centre  of  the  tubes,  when  the  vernier- 
plate  is  turned  around  horizontally;  i.  e.,to  make  the  plane  of 
the  levels  perpendicular  to  the  vertical  axis  of  the  instrument : 

To  test  this,  turn  the  vernier-plai;e  till  each  of  the  plate- 
levels  is  parallel  to  an  opposite  pair  of  the  parallel  plate- 
screws,  and  bring  each  bubble  to  the  middle  of  its  tube,  by 
the  screws  to  which  it  is  parallel.  Then  turn  the  plate  half- 
way around.  If  either  of  the  bubbles  runs  from  the  centre  of 

€/ 

the  tube,  bring  it  half-way  back,  by  raising  or  lowering  one 
end  of  the  tube,  and  the  rest  of  the  way,  by  the  parallel  plate- 
screws.  Again,  turn  the  plate  half-way  around,  and  repeat 
the  operation,  if  necessary.  The  other  tube  must  be  tested, 
and,  if  necessary,  adjusted  in  the  same  way. 

Second  Adjustment.  To  cause  the  line  of  collimation  to 
revolve  in  a  plane ;  i.  e.,  to  make  the  line  of  collimation  per- 
pendicular to  its  axis : 

Set  up  the  instrument  and  level  it  carefully.  Sight  to 
some  well-defined  point,  as  far  off  as  can  be  distinctly  seen. 

FIG.  74. 


Clamp  the  instrument  so  that  there  can  be  no  movement  hori- 
zontally, turn  the  telescope  over,  and  fix  another  point  (as  a  nail 


METHODS  AND  INSTRUMENTS.  59 

driven  in  a  stake)  precisely  in  the  line  of  sight,  and  at  an 
equal  distance  from  the  instrument. 

In  the  figure  let  A  be  the  place  of  the  instrument  and  B  the 
first  point  sighted  to.  If  the  vertical  cross-hair  is  in  adjust- 
ment, the  line  of  sight,  on  turning  over  the  telescope,  will 
strike  at  C,  A  C  being  a  prolongation  of  the  straight  line  A  B. 
If  not  in  adjustment,  it  will  strike  on  one  side,  as  at  D.  E~ow 
loosen  the  clamp,  turn  the  vernier-plate  half-way  around,  and 
-  sight  to  the  first  object  selected.  Again  clamp  the  instru- 
ment and  turn  over  the  telescope.  The  line  of  sight  will  now 
strike  at  E,  as  far  to  the  right  of  the  true  line  as  D  is  to  the 
left. 

To  correct  this,  move  the  vertical  cross-hair  till  the  line  of 
sight  strikes  half-way  between  E  and  C.  Verify  again,  and 
repeat  the  operation,  if  necessary. 

Third  Adjustment.  To  cause  the  line  of  collimation  to 
move  in  a  truly  vertical  plane  when  the  telescope  is  revolved ; 
i.  e.,  to  make  the  axis  of  the  line  of  collimation  parallel  to  the 
plane  of  the  levels  : 

Set  up  the  instrument  near  the  base  of  a  spire,  or  other  high 
object,  and  level  it  carefully.  Sight  to  some  well-defined  point 
on  the  top  of  the  object.  Clamp  the  instrument  so  that  there 
can  be  no  motion  horizontally,  turn  down  the  telescope  and 
fix  a  point  at  the  base  of  the  object,  precisely  in  the  line  of 
sight.  Now  loosen  the  clamp,  turn  the  vernier-plate  half-way 
around,  and  sight  again  to  the  point  on  the  top  of  the  object. 
Again  clamp  the  instrument,  and  turn  down  the  telescope. 
If  in  adjustment,  the  line  of  sight  will  again  strike  the  point 
fixed  at  the  base.  If  not,  the  apparent  error  is  double  the 
real  error.  Make  the  adjustment  by  raising  or  lowering  one 
end  of  the  axis  by  means  of  a  screw,  placed  in  the  standard 
for  that  purpose. 

Fourth  Adjustment.  To  cause  the  line  of  collimation  of 
the  telescope  to  be  horizontal  when  the  bubble  of  the  level 
attached  to  it  is  in  the  centre  of  its  tube : 

The  verification  and  adjustment  is  the  same  as  the  opera- 
tion of  adjusting  the  Y  level  by  the  "  peg  method,"  Art.  (38). 


60        LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

The  operations  of  centring  the  object-glass  and  eye-piece 
are  the  same  as  for  the  level,  Art.  (37). 

Another  adjustment  is  necessary  in  order  that  the  vernier 
of  the  vertical  circle  may  read  zero  when  the  bubble  is  in  the 
centre.  This  is  verified  in  various  ways  : 

1.  By  simple  inspection. 

2.  By  reversion.     Sight  to  some  point.     Note  the  reading 
on  the  vertical  circle.     Turn  the  telescope  half-way  around 
horizontally.     Turn  over  the  telescope  and  again  observe  the 
same  point,  and  note  the  reading.     Half  the  difference  (if 
any)  of  the  two  readings  is  the  error. 

The  principle  is  that  given  in  L.  S.  (334). 
This  method    requires  the  instrument  to  be   a  transit- 
theodolite. 

3.  By  reciprocal  observations.     Observe  successively  from 
each  of  two  points  to  the  other.     Half  the  difference  of  the 
readings  equals  the  index-error. 

When  the  verification  has  been  made,  the  error  may  be 
rectified  on  the  instrument,  or  noted  as  a  correction  to  each 
observation,  when  the  instrument  is  large  and  delicate. 

(94.)  Field- Work.  To  measure  horizontal  angles.  Set  the 
transit  so  that  its  centre  shall  be  precisely  over  the  angular 
point.  This  is  done  by  means  of  a  plumb-line,  suspended 
from  the  centre  of  the  instrument.  Level  the  instrument 
carefully.  Sight  to  a  rod,  held  at  some  point  on  one  of  the 
lines,  as  at  B  in  the  figure  (A  being  the  place  of  the  transit), 
and  note  the  reading.  Then  loosen  the  clamp  of  the  vernier- 
plate,  keeping  the  other  plate  clamped ;  sight  to  a  rod  held  at 
some  point  on  the  second  line,  as  at  C,  and  again  note  the 

FIG.  75. 


reading.     The  difference  of  the  two  readings  will  give  the  an- 
gle BAG.     This  is  the  angle  of  intersection. 


METHODS  AND   INSTRUMENTS  61 

To  measure  the  angle  of  deflection ^  D  A  C,  i.  e.,  the  angle 
between  A  C  and  B  A  -prolonged :  After  sighting  to  B,  turn 
over  the  telescope.  It  will  now  point  toward  D,  in  the  line 
B  A  prolonged.  Note  the  reading,  sight  to  C,  and  again  note 
the  reading.  The  difference  of  the  readings  will  give  the  re- 
quired angle. 

Vertical  angles  are  measured  similarly  to  horizontal  ones, 
only  using  the  vertical  instead  of  the  horizontal  circle. 

Traversing.  In  this  method  of  surveying  and  recording  a 
line,  the  direction  of  each  successive  portion  is  determined, 
not  by  the  angle  which  it  makes  with  the  line  preceding  it, 
but  with  the  first  line  observed,  or  some  other  constant  line. 
The  operation  consists  essentially  in  taking  each  back-sight 
by  the  lower  motion  (which  turns  the  circle  without  changing 
the  reading),  and  taking  each  forward  sight  by  the  upper  mo- 
tion, which  moves  the  vernier  over  the  arc  measuring  the  new 
angle ;  and  thus  adds  it  to  or  subtracts  it  from  the  previous 
reading. 

FIG-  76'  Set  up  the  instrument 

at  some  station,  as  B  ; 
put  the  vernier  at  zero, 
and,  by  the  lower  mo- 
tion, sight  back  to  A. 
Tighten  the  lower  clamp, 
reverse  the  telescope,  loosen  the  upper  clamp,  sight  to  C  by 
the  upper  motion,  and  clamp  the  vernier-plate  again.  Re- 
move the  instrument  to  C,  sight  back  to  B  by  the  lower  motion. 
Then  clamp  below,  reverse  the  telescope,  loosen  the  upper 
clamp,  and  sight  to  D  by  the  upper  motion.  Then  go  to  D 
and  proceed  as  at  0 ;  and  so  on.  The  reading  gives  the  an- 
gles measured  to  the  right  or  "  with  the  sun,"  as  shown  by 
the  arcs  in  the  figure. 

(95.)  Angular  Profiles.  A  section  or  profile  of  a  tolerably 
uniform  slope  is  most  easily  obtained,  as  shown  in  the  figures, 
by  measuring  the  heights  or  depths  below  an  inclined  line, 
instead  of  below  a  level  line. 


62         LEVELLING,  TOPOGRAPHY,  AND   HIGHER   SURVEYING. 

FIG.  77. 


1C 


FIG.  78. 


A  cross-section  for  a  road  may  be  taken  in  the  same  way. 

(96.)  Bunder's  Level.  It  is  a  pear-shaped  instrument,  hav- 
ing two  graduated  circles;  one  vertical,  having  a  weight 
attached  so  as  to  keep  it  in  the 
same  vertical  position  when  in 
use;  and  the  other,  a  horizontal 
graduated  circle,  made  light  and 
carried  around  by  a  magnetic  nee- 
dle, so  that  the  instrument  can  be 
used  as  a  compass  as  well  as  a 
slope  or  angular  level.  It  has  a 

convex-glass,  or  lens,  in  the  smaller  end,  through  which  can 
be  seen  a  hair  which  covers,  on  the  circle,  the  number  of  the 
degrees  of  the  angle  of  inclination,  or  of  the  horizontal  angle. 

The  sights  are  on  the  top  or  sides,  according  as  it  is  used 
as  a  compass  or  slope-measurer.  It  is  used  by  sighting  to  the 
object,  and  at  the  same  time  reading  off  the  angle,  the  hair 
covering  the  zero-mark  when  the  instrument  is  level. 


FIG.  79. 


(97.)  German  Universal  Instrument.  Its  use  is  to  enable  the 
observer  to  sight  to  an  object  nearly  or 
quite  overhead.  It  consists  of  a  tele- 
scope having  the  part  which  carries  the 
eye-piece  at  right  angles  to  the  part 
carrying  the  object-glass,  instead  of  be- 
ing in  the  same  straight  line,  as  in  an 


r^ 


SIMPLE  ANGULAR  LEVELLING. 


63 


ordinary  telescope.  The  part  containing  the  eye-piece  is  con- 
nected with  the  other  part  at  the  axis,  and  is  in  the  same  line 
with  the  axis. 

In  the  telescope  is  placed  a  small  mirror,  or  reflector,  or 
(what  is  still  better)  a  triangular  prism  of  glass,  at  an  angle 
of  45°  to  the  line  of  sight.  Thus  the  observer  can  keep  his 
eye  at  the  same  place  at  any  inclination  of  the  telescope. 


CHAPTEE  II. 


SIMPLE      ANGULAR      LEVELLING. 


A. — For  Short  Distances. 


FIG.  80. 


(98.)  Principle.  For  short  dis- 
tances, curvature  and  refraction 
may  be  neglected.  Thus,  if  the 
height  of  a  wall,  house,  tree,  etc., 
be  desired,  note  the  point  where  the 
horizontal  line  strikes  the  wall,  etc., 
and  add  its  height  above  the  ground 
to  that  calculated  by  the  formula  : 


B  C  =  A  C.  tang.  BAG.  [1.] 

i 
(99.)  The  "best-condition"  angle  for  observation  (see  L.  S., 

383)  is  45°.  Hence,  in  setting  the  instrument,  we  should, 
where  practicable,  have  the  distance  about  equal  to  the  height 
of  the  point  whose  height  we  wish  to  ascertain. 


LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


B.  —  For  Greater  Distances. 

(100.)  Correction  for  Curvature.  AC  is 
the  line  of  apparent  level,  as  given  by  the 
instrument,  and  A  C'  is  the  line  of  true 
level.  Calling  the  angle  A  C  B  =  90° 
(which  it  is  approximately  for  moderately 
great  distances),  formula  [1]  gives  B  C  as 
the  height  of  B  above  A.  But  B  C'  is  the 
true  difference  of  heights  of  A  and  B. 

A  correction  for  the  curvature  of  the 
earth  must  therefore  be  made.  It  may 
be  done  in  two  ways  :  either  by  calcu- 
lating C  C',  and  adding  it  to  B  C,  obtained 
by  formula  [1]  ;  or  by  calculating  the  an- 
gle C  A  C',  adding  it  to  B  A  C,  and  then 
applying  the  formula  [1]  to  the  angle 
B  AC'. 


(101.)  Correcting  the  Result. 
we  have,  by  (14)  : 


Expressing  the  distance  by 


=  0.000000023936>P. 

Then,  calling  A  C  B  a  right  angle,  we  have ; 

B  C'  =  J&  x  tang.  B  A  C  +  0.000000023936^  in  ft.  [2.] 

The  arc  A  C'  and  the  straight  lines  A  C'  and  A  C  are  all 
three  approximately  equal. 

(102.)  Correcting  the  Angle.    The  angle  C  A  C'  =  i  A  O  C', 

the  central  angle,  which  is  measured  by  the  arc  A  C'  or  Tc. 

The  length  of  the  arc  subtending  one  minute 

2?r  x  20888629 


SIMPLE  ANGULAR  LEVELLING. 

Then,  for  any  arc,  &,  the  angle  O  in  minutes 
Jc 


6076 


=  0.0001646& ; 


and  the  angle  C  A  C'  (in  minutes)  =  O.OOOOS23&. 

Adding  this  to  the  observed  angle,  BAG,  and  calling 
A  C'  B  a  right  angle,  we  have,  by  [1]  : 


B  C'  =  k    tang.  (B  A  C  +  0.0000823&). 


[3.] 


Fio.  82. 


(103.)   Correction  for  Refraction.     The  effect  of  refraction 

causes  the  angle  actually  ob- 
served to  be,  not  CAB,  but 
C  A  B',  which  will  be  desig- 
nated by  a°.  For  small  dis- 
tances, B  and  B'  sensibly  co- 
incide. The  correction  for 
refraction  may  be  made  in 
two  ways,  as  for  curvature. 

To  correct  the  result  ~by 
finding  B  B'.  It  varies  very 
irregularly,  with  wind,  ba- 
rometer, temperature,  etc. ; 
but  is  usually  taken,  as  an 
average,  B  B'  =  0.16  C  C'. 

Subtracting  this  from  the  value  of  B  C',  in  formula  [2],  it 
becomes  B  C'  =  k .  tang.  B'A  C  +  0.00000002&8.  [4] 

To  correct  the  observed  angle.     Subtract  from  it  the  angle. 
B  A  B',  which  is  about  0.16  of  the  angle  C  A  C'. 

This  changes  formula  [3]  to 

B  C'  =  TG  .  tang.  (B'  A  C  +  0.000069&).  [5.] 

C. — For  Very  Great  Distances. 

(104.)  Correction  for  Curvature.     As  before,  there  are  two 
methods  of  making  the  correction. 
.5 


66         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

For  these  distances  we  cannot  consider  the  angle  at  C'  a 
right  angle.     The  triangle  ABC  gives 


sin. 

To  find  the  angle  B,  we  have,  in  the  triangle  B  A  O, 
B  =  180°  -  (O  +  B  A  0), 
B  =  180°  -  (O  +  90°  +  B  AC), 
B=    90°-(O  +  BAC); 
Hence,  sin.  B  =  cos.  (O  +  B  A  C). 

mi.         T}  /-i         7  sin-  B  A  C 

Then,BC^.—  (Q  +  BAC), 

and  B  0'=  B  C  +  C  C'=  Tc.  ~^^^C}  +  0.000000023936&1 

+  0.00000008898W. 


Correcting  the  Angle.  In  the  triangle  A  B  C',  getting 
expressions  for  the  angles,  and  using  the  sine  proportion,  as 
before,  in  A  B  C,  we  have  : 

sin.  (BAG -HO), 
^'cos.  (BAC+0) 

B0/  =  .^.(B  A  0  +  0.0000823*).  pj 


(105.)  Correction  for  Refraction.    Formula  [6]  becomes 

sin.  (B^AC-  0.00001316^) 
-  *'        (B'A  C  +  +  0' 


SIMPLE  ANGULAR  LEVELLING.  67 

Formula  [7]  becomes,  diminishing  B  A  C  in  both  numer- 
ator and  denominator  by  0.08  of  O, 


B  °'  = 


sin.  (B'AC  +  0.00006913^) 
cos.  (B7  A  C  +  0.0001514:3^) 


[9.] 


FIG.  83. 


(106,)  Reciprocal  Observations  for  Cancelling  Refraction.  Ob- 
serve the  reciprocal  angles  of  eleva- 
tion and  depression  from  each  point 
to  the  other.  Call  these  angles 
a  and  0.  Then:- 


cos. 


. 
0) 


[10.] 


- 


NOTE.  —  Angle  0,  in  minutes  =  0.0001646&. 
Log.  0.0001646  =T.2164298. 

When  zenith  distances  are  ob- 
served, they  are  denoted  by  6  and  <5'. 
Then  formula  [10]  becomes  : 


[10'.] 


When  O  is  very  small,  compared  with  the  other  angles,  by 
neglecting  it  we  have : 


[11.] 


Using  zenith  distances,  this  becomes :     • 
B  C'  =  JG  .  tang.  J  (<P  —  cJ). 


(107.)  Reduction  to  the  Summits  of  the  Signals.  Stations  a 
and  ~b  cannot  be  seen  from  one  another.  Signals,  Art.  (240),  are 
erected  at  each  point,  and  from  a  the  angle  B  a  C  =  A  is 
observed ;  and  from  I  the  angle  A  5  D  =  B.  The  heights  of 
the  signals  above  the  instrument  are  h  and  h'. 


68         LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


Kequired  the  reduced  angles  a  and/?. 


Fro.  84. 


h .  cos.  A 

a  =  A  ~  F.1ETF 

hf.  cos.  B 


.  sn. 


[12]. 


The  difference  is  in  seconds. 

Usually,  in  such  cases,  zenith  dis- 
tances are  taken,  and  the  observed  an- 
gles are  called  A  and  A'.  The  re- 
duced angles  are  6  and  d'.  Then  the  formulas  become : 


The  difference  is  in  seconds. 

Instead  of  h  and  h'-  some  writers  use  dR  and  dR'  ;  or  dA 
and  dA.',  meaning  difference  of  height,  and  difference  of  alti- 
tude. 

For  great  exactness,  instead  of  using  the  mean  radius  of 
the  earth  to  get  O,  the  radius  at  the  point  of  observation 
is  used. 

(108.)  When  the  height  of  the  signal  above  the  instrument 
cannot  be  measured,  if  the  signal  be  conical,  like  a  spire,  etc., 

FIG.  85. 


to  find  B  B'  we  measure  two  diameters,  2  E  and  2  r,  and  the 
distance  apart,  h. 


SIMPLE  ANGULAR  LEVELLING.  £9 

Then,  BB'  =  j^^.  [14.] 

If  the  oblique  distance  I  be  measured  instead  of  A,  then 
R      


When  the  spire  is  very  acute,  then  this  method  is  inac- 
curate. 


FIG.  86. 


O  V.     f'' 

V*  V — 


Take  -some  point,  A,  and  observe  zenith  distances,  d,  <J", 
and  6'.     Then : 


. 
" 


[16.] 


(109.)  Levelling  by  the  Horizon  of  the  Sea.  From  an  emi- 
nence, as  B,  sight  to  the  sea  horizon,  and  measure  d°=  angle 
ABZ.  Then: 

FIG.  87. 

<?3 
--*3s 


B(y=iB|j^J  (rf°-90°)a  [1+i  (j~)  (^°-900)2].  [17.] 

(d°  —  90°)  is  to  be  reduced  to  seconds.  It  is  equal  to  the 
angle  of  depression  at  B ;  n  is  the  coefficient  of  refraction.  It 
is  taken  at  0.08. 


70          LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


CHAPTER  III. 

COMPOUND  ANGULAR  LEVELLING-. 

THE  following  problems  may  mostly  be  reduced  to  a  com- 
bination of:  first,  determining  the  inaccessible  distance  to  a 
point  immediately  under  (or  over)  the  point  whose  height  is 
desired,  and  then  using  this  distance  to  obtain  that  height. 

FIG.  88. 


(110.)  By  Angular  Co-ordinates  in  one  Plane.  Take  two  sta- 
tions, A  and  D,  in  the  same  vertical  plane  with  B.  At  A 
observe  the  angles  of  elevation  of  B  and  D.  Measure  A  D. 
At  D  observe  angle  A  D  B.  Then,  in  triangle  A  B  D  we  get 
A  B,  and  in  triangle  B  A  C  we  get  B  C. 


sin.  BD  A.  sin.  B  AC 
sin.  A  B  D 


[18.] 


For  great  distances,  the  corrections  for  curvature  and  re- 
fraction are  to  be  made  as  in  last  chapter. 

If  A  D  be  horizontal,  the  same  formula 
applies ;  but  there  is  one  angle  less  to  meas- 
ure ;  since  B  A  C  =  B  A  D.  Formula  [18] 
gives  the  height  of  B  above  A. 


FIG.  89. 


' 


If  the  height  of  B  above  D,  in  Fig.  88,  be 
'desired,  find  B  D  in  the  triangle  BAD,  observe  the  angle  of 
elevation  of  B  from  D,  and  then  the  desired  height  equals 


B  D.  sin.  B  D  E. 


COMPOUND  ANGULAR  LEVELLING. 


71 


Otherwise,  find  height  of  D  above  A,  and  subtract  it 
from  B  C. 

(111.)  By  Angular  Co-ordinates  in  several  Planes.  On  irreg- 
ular ground,  when  the  distance  between  the  two  points  is 
unknown,  the  operations  for  finding  it  by  the  various  methods 
of  L.  S.,  Part  YIL,  Chapter  III.,  may  be  combined  with  the 

observation  of  vertical  angles,  thus : 

• 

FIG.  90. 


At  A  measure  the  vertical  angle  of  elevation,  B  A  C.    Also 
measure  the  horizontal  angle,  C  A  D,  to  some  point,  D,  and 
measure  horizontally  the  distance,  AD.     At  D  measure  the  . 
horizontal  angle  ADC.     Then  : 


=  AC.tang.BAC  = 


B"1-  ADC.  tang.  B  A  C 

i T — TV^pr 

sin.  A  C  D 


[19.] 


FIG.  91. 


(112.)  Conversely.  The  distance 
may  be  obtained  when  the  height 
is  known. 

Let    C  B  be  a  known  height : 
Then,   AC  =  CB. tan.  ABC. 


72         LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


FIG.  92. 


B  C  is  a  known  height,  and  D  E  an 
inaccessible  line  in  the  same  horizontal 
plane  as  C.  Find  CD  and  CE  by 
the  last  method,  and  measure  the  hor- 
izontal angle  E  C  D  subtended  at  C  by 
ED. 

Then  two  sides  and  the  included 
angle  of  a  triangle  are  known,  to  find  the  third  side. 


PART  III. 
BAROMETRIC    LEVELLING 


CHAPTEE  I. 

PRINCIPLES     AND     FORMULAS. 

•  (113.)  Principles.  The  difference  of  the  heights  of  two 
places  may  be  determined  by  finding  the  difference  of  their 
depths  below  the  top  of  the  atmosphere  in  the  same  way  as 
the  comparative  heights  of  ground  under  water  are  determined 
by  the  difference  of  the  depths  below  the  top  of  the  water. 
The  desired  height  of  the  atmosphere  above  any  point,  such 
as  the  top  of  a  mountain,  or  the  bottom  of  a  valley,  is  deter- 
mined by  weighing  it.  This  is  done  by  trying  how  high  a 
column  of  mercury  or  other  liquid  the  column  of  air  above  it 
1  will  balance ;  or  what  pressure  it  will  exert  against  an  elas- 
tic box  containing  a  vacuum,  etc.,  etc.  Such  instruments  are 
called  Barometers. 

(114.)  Applications.  Since  the  column  of  mercury  in  the 
barometer  is  supported  by  the  column  of  air  above  it,  the 
mercury  sinks  when  the  barometer  is  carried  higher,  and  vice 
versa. 

The  weight  of  any  portion  of  air  decreases  from  the  surface 
of  the  earth  to  the  assumed  surface  of  the  atmosphere.  It  has 
been  found  that,  as  the  heights  to  which  the  barometer  is  car- 
ried increase  in  arithmetical  progression,  the  weights  of  the 
column  of  air  above  the  barometer,  and  consequently  its  read- 


^4->iXM*< 

^    74          LEVELLING,  TOPOGRAPHY,  A.SV  HIGHER  SURVEYING. 

ings,  decrease  in  geometrical  progression.  Consequently,  the 
difference  of  the  heights  of  any  two,  not  very  distant,  points 
on  the  earth's  surface,  is  proportional  to  the  difference  of  the 

X  logarithms  of  the  readings  of  the  barometer  at  those  points ; 
i.  e.,  equal  to  this  latter  difference  multiplied  by  some  constant 

^  coefficient.     This  is  found  by  experiment  to  be  60159,  at  the 

^  freezing  point,  or  temperature  of  32°  F.,  the  readings  of  the 

mercury  being  in  inches,  and  the  product,  which  is  the  differ- 

f  ence  of  height,  being  in  feet. 

^       Several  corrections  are  necessary. 

fc        (115.)  Correction  for  Temperature  of  the  Mercury.     If  the 

\  temperature  of  the  mercury  be  different   at  the    two    sta- 
"  tions,  it  is  expanded  at  the  one,  and  contracted  at  the  other, 
to  a  height  different  from  that  which  is  due  to  the  mere 
"\  weight  of  the  air  above  it. 

Mercury  expands  about  10ooo  of  its  bulk  for  each  degrfee 
of  F.  Therefore,  this  fraction  of  the  height  of  the  column  is 
to  be  added  to  the  height  of  the  colder  column,  or  subtracted 
from  the  height  of  the  warmer  one,  in  order  to  reduce  them 
to  the  same  standard.  A  thermometer  is  therefore  attached 
to  the  instrument  in  such  a  manner  as  to  give  the  temperature 
of  the  mercury. 

If  a  brass  scale  is  used,  the  correction  is  1 0  090  0  0  for  each 
,     degree  F. 

^ 

(116.)  Correction  for  Temperature  of  the  Air.  The  warmer 
the  air  is,  the  lighter  it  is  ;  so  that  a  column  of  warm  air  of 
*"£  any  height  will  weigh  less  than  when  it  is  colder.  Con- 
sequently, the  mercury  in  warm  air  falls  less  in  ascending  any 
height,  and  is  higher  at  the  place  than  it  otherwise  would  be. 
Hence  the  height  given  by  the  preceding  approximate  result 
will  be  too  small,  and  must  be  increased  by  ^y  part  for  each 


J       degree  F.  that  the  temperature  of  the  air  is  above  32°.     The 
effect  of  moisture  in  the  air  changes  this  fraction  to  ^-J-g-. 
* 

(117.)  Other  Corrections.    For    very  great    accuracy,   we 
should  allow  for  the  variation  of  gravity,  corresponding  to  the 


PRINCIPLES  AND   FORMULAS. 


75 


variation  of  latitude  on  either  side  of  the  mean.  So,  too,  we 
should  allow  for  the  decrease  of  gravity  corresponding  to  any 
increase  of  height  of  the  place. 

(118.)  Rules  for  Calculating  Heights  by  the  Mercurial  Ba- 
rometer. 1.  At  each  station  read  the  barometer;  note  its 
temperature  by  the  attached  thermometer,  and  note  the  tem- 
perature of  the  air  by  a  detached  thermometer. 

2.  Multiply  the  height  of  the  upper  column  by  the  differ- 
ence  of  readings  of  the  attached  thermometer,  and  that  by 
i  o  o9o  o  o  ?  and  add  the  product  to  the  upper  column,  if  that  be 
the  colder,  or  subtract  it,  if  that  be  the  warmer.     This  gives 
the  corrected  height  of  the  mercury. 

3.  Multiply  the  difference  of  the  logarithms  of  the  cor- 
rected heights  of  the  mercury  (i.  e.,  the  corrected  upper  one 
and  the  lower  one)  by  60159,  and  the  product  is  the  approxi- 
mate difference  of  heights  of  the  places  in  feet  for  the  temper- 
ature of  32°. 

4.  Subtract  32°  from  the  arithmetical  mean  of  the  temper- 
atures of  the  detached  thermometer;  multiply  the   approxi- 
mate altitude  by  this  difference ;  divide  the  product  by  450  ; 
add  the  quotient  to  the  approximate  altitude,  and  the  sum  is 
the  corrected  altitude. 


(119.)  Formulas. 

in  formulas,  thus : 


The  rules  just  given  are  best  expressed 


At  Lower  Station. 

At  Upper  Station. 

H 

h' 

Temperature  of  Mercury                          .  . 

T 

T' 

Temperature  of  Air  

t 

f 

Calling  the  reduced  height  of  mercury  at  upper  station  A, 
we  have,  by  Eule  2 : 

A  =  A' -f  0.00009  (T  —  T')  A'.  [1.] 

(K  B.    If  T  is  more  than  T,  the  product  will  be  sub- 
tractive.) 


76         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

Then,  by  Rule  3,  we  have : 

Approx.  height  =  60159  (log.  H  —  log.  h). 
By  Rule  4,  the  correction  for  temperature  of  air 

t'  -  64 


=  approx.  height  x 


900 


Adding  this  correction  to  the  approximate  height,  and  fac- 
toring the  sum,  we  get  : 


Corrected  ht.  =  60159(log.  H  -  log.  A)  (l  -f  -     -gojp        [2]- 


(120.)  To  Correct  for  Latitude.  Multiply  the  preceding  re- 
sult by  0.00265  .  cos.  2L  (L  being  the  latitude),  and  add  (alge- 
braically) the  product  to  the  preceding  result. 

At  45°,  correction  is  zero.  At  equator  it  is  +  0.00265.  At 
pole  it  is  —0.00265. 

To  Correct  for  Elevation  of  the  Place.  Call  the  last  cor- 
rected height  #',  and  the  height  of  the  lower  place  above  the 
level  of  the  sea  S,  and  add  to  x'  this  quantity  : 

xf  +  52251  S 


20888629        10444315 

(121.)  Final  English  Formula.     Combining  the  previous  re- 
sults into  one  formula,  we  get : 


t.= 60159  (log.  H-log.  Ji) 


900 


'). 


(1  +  0.00265.  cos.  2  L), 


52251  S 


20888629 


\ 
10444315; 


[3]- 
In  this  formula,  the  three  quantities  under  each  other  are 

three  factors. 


PRINCIPLES  AND   FORMULAS.  77 

Usually,  only  the  first  factor  is  required,  and  then  we  have 
formula  [2].  Using  the  second  also  we  correct  for  latitude ; 
and  using  the  third,  for  the  elevation. 

(122.)  French  Formulas.  French  barometers  are  graduated 
in  French  millimetres,,  each  =  0.03937  inch.,  and  the  ther- 
mometer is  centigrade,  in  which  the  freezing-point  is  zero, 
and  boiling-point  100°  : 

a°Cent.  =(|a  +  32)°  F. 

Then,  the  French  formula  corresponding  to  [3]  is  the  fol- 
lowing (H  and  A'  being  in  millimetres,  and  the  temperatures 
centigrade) : 


/        T  —  T\ 

h = h>  I1  +  -6200-;- 

And  the  difference  of  heights  in  metres 

/        2(*+0\ 
V1  +  "1000~VJ 


•=  18336  (log.  H-  log.  h) 


(1  +  0.00265.  cos.  2  L), 


/ 
V 


a/.'+  15926        _S  _ 
6366198  ~+  3183099 


(123.)   Babinet's  Simplified   Formula,   without  Logarithms. 

—  T 


—      \ 

hr  is  reduced  to  A,  as  before,  viz.  :  h  =  h'  \\  -\  --  goTjTT  )' 

Then,  the  difference  of  heights  in  metres 


The  heights  are  in  millimetres  and  the  temperatures  centi- 
grade. 


78          LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

Example.     H  =  755.      h  =  745 
t  =  15°        t'  =  10° 

.  Ht.  =  16000  l  +  =  112m. 


Correct  result  is  111.6  m. 

This  formula  is  a  very  near  approximation  for  moderate 
heights. 

Babinet's  formula  in  English  measures  (the  heights  being 
in  inches,  and  temperatures  Fahrenheit)  is  in  feet  : 


52494    =r- — T 


(124.)  Tables.  These  shorten  the  operations  greatly.  The 
most  portable  are  in  "  Annuaire  du  Bureau  des  Longitudes." 
The  most  complete  are  Prof.  Guyot's,  published  by  the  Smith- 
sonian Institute  at  Washington. 

(125.)  Approximations.  TL-  of  an  inch  difference  of  read- 
ings at  two  places  corresponds  to  about  90  feet  difference  of 
elevation.  1  millimetre  difference  of  readings  corresponds  to 
about  10J  metres  difference  of  height,  or  about  34  feet. 

This  is  correct  near  the  freezing-point,  and  near  the  level 
of  the  sea.  The  height  corresponding  to  any  given  difference 
of  readings  increases,  however,  with  the  temperature  and  with 
the  height  of  the  station.  Thus,  at  70°  F.,  -fa  of  an  inch  cor- 
responds to  an  elevation  of  95  feet ;  and  1  mm.  at  30°  Cent, 
corresponds  to  llf  metres,  or  about  40  feet. 


INSTRUMENTS.  79 

CHAPTER   II. 

INSTRUMENTS. 

(126.)  BAROMETERS  made  for  levelling  are  called  Mountain 
Barometers.     They  are  either  cistern  barometers  or  siphon 
barometers.     The  best  of  the  former  is  Fortin's,  as  improved 
by  Prof.  Guyot.     (See  Fig.  93.)     This  consists  of  a  col- 
'IG'93'  umn  of  mercury  contained  in  a  glass  tube,  whose  lower 
end  is  placed  in  a  cistern  of  mercury.    The  tube  is  cov- 
ered with  a  brass  case,  terminating  at  the  top  in  a  ring, 
A,  for*  suspension,  and  at  the  bottom  in  a  flange,  B,  to 
which  the  cistern  is  attached. 

At  C  is  a  vernier  by  which  the  height  of  the  mercury 
»D  is  read  off.  The  zero  of  the  scale  is  a  small  ivory  point, 
P.  The  mercury  in  the  cistern  is  raised  or  low-  Flo  94 
ered,  by  means  of  the  milled-headed  screw  O,  till 
its  surface  is  just  in  contact  with  the  ivory  pBint. 
At  E  is  the  attached  thermometer  which  indi- 
cates the  temperature  of  the  mercury.  When  it 
is  carried,  the  mercury  is  screwed  up  to  prevent 
breaking  the  glass.1 

In  the  siphon  barometer,  the  cistern  is  dis- 
pensed with.  The  tube  is  turned  up  at  the 
lower  end,  as  shown  in  Fig.  94,  and  a  small  hole, 
at  T,  admits  the  air.  The  difference  of  heights 
of  the  mercury  in  the  two  branches  of  the  tube 
is  taken  as  the  height  of  the  mercurial  column. 
It  is  enclosed  in  a  brass  case,  and  furnished  with  ver- 
niers, thermometers,  etc.,  as  in  the  preceding  form.  It 
is  carried  inverted,  to  avoid  breaking. 

The  best  is  Gay-Lussac's,  improved  by  Bunten. 

1  For  a  complete  description,  see  Tenth  Annual  Report  of  Smithsonian  Insti- 
tute. 


80         LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


(127.)  The  Aneroid  Barometer.  This  is  a  thin  box  of  corru- 
gated copper,  exhausted  of  air.  When  the  air  grows  heavier, 
the  box  is  compressed ;  and  when  the  air  grows  lighter  it  is 


Fro.  95 


expanded  by  a  spring  inside.  This  motion  is  communicated 
by  suitable  levers  to  the  index-hand,  on  the  face,  which  indi- 
cates the  pressure  of  the  atmosphere,  the  face  being  graduated 
to  correspond  with  a  common  barometer. 

It  is  much  used  on  account  of  its  portability,  but  is  not  as 
reliable  as  the  mercurial  barometer. 


(128.)  The  temperature  at 
which  water  boils  varies  with 
the  pressure  of  the  atmos- 
phere, and  therefore  decreas- 
es in  ascending  heights.  Then 
a  thermometer  becomes  a  sub- 
stitute for  a  barometer. 

Approximately,  each  de- 


Temperature 
of  Boiling  Water. 

Corresponding 
Barometer  Readings. 

213° 

30".522 

212° 

29".922 

211° 

29".331 

210° 

28".751 

209° 

28M80 

208° 
• 

2T.618 

INSTRUMENTS. 


81 


gree  of  difference  (Fahr.)  corresponds  to  about  550  feet  differ- 
ence of  elevation,  subject  to  the  usual  barometric  corrections 
for  the  temperature  of  the  air.  For  minute  tables,  see  Guyot's. 

(129.)  Accuracy  of  Barometric  Observations.  This  increases 
with  the  number  of  repetitions  of  them,  the  mean  being 
taken.  With  great  skill  and  experience  they  may  be  depended 
upon  to  a  very  few  feet, 


n- 


PROFESSOR   GUYOT  S   RESULTS. 


Heights  found  by  the 
Barometer. 

Corresponding 
Heights  found  by  the 
Spirit-Level.  " 

6707  feet. 
2752     " 
6291     " 

6711  feet. 

2752    " 
(  6285    " 
(  6293    " 

(130.)  The  observations  at  the  two  places,  whose  difference 
of  heights  is  to  be  determined,  should  be  taken  simultaneously 
at  a  series  of  intervals  previously  agreed  upon,  the  distance 
apart  of  the  places  being  as  short  as  possible.  Distant  places 
should  be  connected  by  a  series  of  intermediate  ones. 


PART  IV. 
TOPOGRAPHY. 


INTKODUCTIOK 

(131.)  Definition,  Topography  is  the  complete  determina- 
tion and  representation  of  any  portion  of  the  surface  of  the 
earth,  embracing  the  relative  position  and  heights  of  its  ine- 
qualities ;  its  hills  and  hollows,  its  ridges  and  valleys,  level 
plains,  slopes,  etc.,  telling  precisely  where  any  point  is,  and 
how  high  it  is. 

It  therefore  determines  the  three  coordinates  of  any  point ; 
the  horizontal  ones  by  surveying,  and  the  vertical  one  by 
levelling. 

The  results  of  these  determinations  are  represented  in  a 
conventional  manner,  which  is  called  "topographical  map- 
ping." 

The  difficulty  ils,  that  we  see  hills  and  hollows  in  elevation, 
while  we  have  to  represent  them  in  plan. 

(132.)  Systems.     Hills  are  represented  by  various  systems : 

1.  By  level  contour-lines,  or  horizontal  sections. 

2.  By  lines  of  greatest  slope,  perpendicular  to  the  former. 

3.  By  shades  from  vertical  light. 

4.  By  shades  from  oblique  light. 

The  most  usual  method  is  a  combination  of  the  first,  second, 
and  third  systems. 


FIRST   SYSTEM. 


83 


CHAPTEE  I. 


FI'BST  SYSTEM. 


BY     HORIZONTAL     C O N T O UB -L  INE  S  . 

(133.)  General  Ideas.  Imagine  a  hill  to  be  sliced  off  by  a 
number  of  equidistant  horizontal  planes,  and  their  intersec- 
tions with  it  to  be  drawn  as  they  would  be  seen  from  above, 
or  horizontally  projected  on  the  map,  as  in  Fig.  96.  These 
are  "contour-lines." 

FIG.  96. 


They  are  the  same  lines  as  would  be  formed  by  water  sur- 
rounding the  hill,  and  rising  one  foot  at  a  time  (or  any  other 
height),  till  it  reached  the  top  of  the  hill.  The  edge  of  the 


FIG.  97. 


FIG.  98. 


FIG.  99. 


84:         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

water,  or*  its  shore,  at  each  successive  rise,  would  be  one  of 
these  horizontal  contour-lines.  It  is  plain  that  their  nearness 
or  distance  on  the  map  would  indicate  the  steepness  or  gentle- 
ness of  the  slopes.  A  right  cone  would  thus  be  represented 
by  a  series  of  concentric  circles,  as  in  Fig.  97  ;  an  oblique  cone, 
by  circles  not  concentric,  bat  nearer  to  each  other  on  the  steep 
side  than  on  the  other,  as  in  Fig.  98 ;  and  by  a  half-egg,  some- 
what as  in  Fig.  99. 

(134.)  Plane  of  Reference.  The  horizontal  plane  on  which 
the  contour-lines  are  projected,  and  to  which  they  are  re- 
ferred, is  called  the  "  plane  of  reference."  This  plane  may  be 
assumed  in  any  position,  and  the  distance  of  the  contour-lines 
above  or  below  it  is  noted  on  them.  It  is  usually  best  to 
assume  the  position  of  the  plane  of  reference  lower  than  any 
point  to  be  represented ;  so  that  all  the  contour-lines  will  be 
above  it,  and  none  of  them  have  minus  signs.  (See  Art.  59.) 

(135.)  Vertical  Distances  of  the  Horizontal  Sections.  These 
depend  on  the  object  of  the  survey,  the  population  of  the  coun- 
try, the  irregularity  of  the  surface,  and  the  scale  of  the  map. 
In  mountainous  districts  they  may  be  100  feet  apart.  On  the 
United  States  Coast  Survey  they  are  20  feet.  For  engineering 
purposes,  5  feet,  or  less.  One  rule  is  to  make  the  distance  in 
feet  equal  to  the  denominator  of  the  ratio  of  the  scale  of  the 
map,  divided  by  600. 

(136.)  Methods  for  determining  Contour-Lines.  They  are  of 
two  classes :  1.  Determining  them  on  the  ground  at  once  ;  2. 
Determining  the  highest  and  lowest  points,  and  thence  de- 
ducing the  contour-lines. 

FIRST  METHOD. 

(137.)  General  Method.  Determine  one  point  at  the  desired 
height  of  one  line,  as  in  Art.  (81)  ;  and  then  "  locate  "  a  line 
at  that  level,  as  in  Art.  (84). 

The  "  reflected  hand-level,"  or  "  reflecting-level,"  or  "  wa- 


FIRST  SYSTEM. 


85 


ter-level,"  are  sufficiently  accurate  between  "bench-marks" 
not  very  distant. 

One  such  line  having  been  determined,  a  point  in  the  next 
higher  or  the  next  lower  one  is  fixed,  and  the  preceding  oper- 
ations repeated. 

(138.)  On  a  long,  narrow  Strip  of  Ground,  such  as  that  re- 
quired for  locating  a  road :  Run  a  section  across  it  at  every  J 
or  -J-  mile,  about  in  the  line  of  greatest  slope.  Set  stakes  on 
these  sections  at  the  heights  of  the  desired  contour-lines, 
and  then  get  intermediate  points  at  these  heights  between  the 
stakes.  These  sections  check  the  levels. 

FIG.  100. 


(139.)  On  a  Broad  Surface.  Level  around  it  setting-stakes, 
at  points  of  the  desired  height,  and  then  run  sections  across  it, 
and  from  them  obtain  the  contour-lines  as  before. 

The  external  lines  here  serve  as  checks  to  the  cross-lines. 

(140.)  Surveying  the  Contour-Lines.  The  contour-lines  thus 
found  may  be  surveyed  by  any  method.  If  they  are  long,  and 
not  very  much  curved,  the  compass  and  chain  and  the  method 
of  "  progression  "  is  best.  (See  L.  S.  220.)  If  they  are  curved 
irregularly,  the  method  of  radiation  is  best.  When  straight 
lines  exist  among  them,  such  as  fences,  etc.,  or  can  conveniently 
be  established,  then  rectangular  coordinates  are  most  con- 
venient. 


86        LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

(141.)  Contouring  with  the  Plane-Table,1  It  is  used  to  map 
the  points  as  soon  as  obtained,  thus :  Range  out  an  approxi- 
mately level  line,  and  on  it  set  equidistant  stakes.  At  these 
Btakes  range  out  perpendiculars  to  the  line,  and  set  up  several 
stakes  on  them  for  the  alignment  of  the  rodman.  Draw  these 
lines  on  the  plane-table.  Set  up  and  "  orient "  (L.  S.  456) 
the  table  on  the  ground.  Send  the  rod  along  one  of  the  per- 
pendiculars till  it  comes  to  a  point  of  the  right  height.  Then 
sight  to  it  with  the  alidade,  and  its  edge  will- cut  the  corre- 
sponding line  on  the  table  at  the  correct  place  on  the  plat.  So 
for  the  other  perpendiculars. 

SECOND  METHOD. 

(142.)  General  Nature,  This  method  consists  in  determin- 
ing the  heights  and  positions  of  the  principal  points,  where 
the  surface  of  the  ground  changes  its  slope  in  degree  or  in 
direction,  i.  e.,  determining  all  the  highest  and  lowest  points 
and  lines,  the  tops  of  the  hills  and  bottoms  of  the  hollows, 
ridges  and  valleys,  etc.,  and  then,  by  proportion  or  interpola- 
tion, obtaining  the  places  of  the  points  which  are  at  the  same 
desired  level.  The  heights  of  the  principal  points  are  found 
by  common  levelling,  and  their  places  fixed  as  in  Art.  (141). 

The  first  method  is  more  accurate.  The  second  is  more 
rapid. 

(143.)  Irregular  Ground.  When  the  ground  has  no  very 
marked  features,  run  lines  across  it  in  various  directions,  and 
level  along  them,  taking  heights  at  each  change  of  slope,  just 
as  in  taking  sections  for  profiles. 

Otherwise,  thus :  Set  stakes  on  four  sides  of  the  field,  so 
as  to  enclose  it  in  a  rectangle,  if  possible,  as  in  Fig.  101. 
Place  the  stakes  equidistant,  so  that  the  imaginary  visual 
lines  connecting  them  would  divide  the  surface  into  rectangles. 
Send  the  rod  along  one  of  these  lines  till  it  gets  in  the  range 
of  a  cross-one,  and  observe  to  it  there.  Put  down  the  ob- 
served heights  of  these  points  at  the  corresponding  points  on 

1  For  description  and  method  of  using  the  Plane-table,  see  L.  S.  Part  VIII. 


FIRST   SYSTEM. 


87 


the  plat,  on  which  these  lines  have  been  drawn.     The  con- 
tour-lines are  determined  as  in  Art.  (146). 


FIG.  101. 


(144.)  On  a  Single  Hill.  Proceed  thus :  From  its  top,  range 
lines  down  the  hill,  in  various  directions,  and  take  their  bear- 
ings. Set  stakes  on  them  at  each  change  of  slope,  and  note 
the  heights  and  distances  of  these  stakes  from  the  starting- 
point,  and  plat  their  places.  The  contour-lines  are  then  put 
in  as  in  Art.  (146). 

"With  a  transit,  the  heights  of  the  points  could  be  deter- 
mined by  vertical  angles ;  and  also  their  distances  with  stadia- 
hairs,  their  directions  being  given  by  the  horizontal  circle  of 
the  transit.  The  French  use  for  this  purpose  a  "  levelling 
compass." 

(145.)  For  an  Extensive  Topographical  Survey.  Proceed  thus : 
Set  up,  and  get  the  height  of  the  cross-hairs  from  some  bench- 
mark, and  get  the  heights  of  high  and  low  prominent  points 
all  around.  Then  go  beyond  these  points  and  set  up  again. 
Sight  to  one  of  these  known  points  as  a  "  turning-point,"  and 
get  the  heights  of  all  the  points  now  in  sight,  as  before.  Then 
go  beyond  these  again,  and  so  on.  The  places  of  these  new 
points  are  fixed  as  before. 


88         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


(146.)  Interpolation.     The  heights  and  the  places  of  the 
principal    points    being    determined,  by   either  of  the   pre- 


50.00 


Fio.  102. 

.45.00  40.00 


35JOO 


35.00 


36.00 


^3300. 


ceding  methods,  points  of  any  intermediate  height,  corre- 
sponding to  any  desired  contour-curve,  are  obtained  by  pro- 
portion. 

If,  in  Fig.  102,  the  heights  of  the  intersection  of  the  lines 
being  found,  as  in  Art.  (143),  and  their  distance  apart  being 
100  feet,  it  is  required  to  construct  contour-curves  whose  dif- 
ference of  heights  is  5  feet :  Taking  for  example  the  one 
whose  height  is  45  feet,  we  see  it  must  fall  between  the  points 
A  and  B,  whose  heights  are  50  feet  and  35  feet ;  and  its  dis- 
tance from  A  will,  be  found  by  the  proportion,  as  15  is  to  5  so 
is  100  to  the  required  distance.  So  on  for  any  number  of 
points.  To  save  the  labor  of  continually  calculating  the  fourth 
proportional,  a  scale  of  proportion  may  be  constructed. 

(147.)  Interpolating  with  the  Sector.  (L.  S.  52.)  This  is 
the  easiest  way.  The  problem  is :  having  given  on  a  plat  two 
points  of  known  height,  to  interpolate  between  them  a  point 
of  any  desired  intermediate  height. 

Take  in  the  dividers  the  distance  between  the  given  points 
on  the  plat ;  open  the  sector  so  that  this  distance  shall  just 


FIRST  SYSTEM. 

reach  between  numbers,  on  the  scale 
marked  L,  corresponding  to  the  dif- 
ference of  the  heights  of  the  two 
given  points;  i.  e.,  from  6  to  6,  or 
7  to  7,  and  so  on.  The  sector  is 
then  set  for  all  the  interpolations 
between  these  two  points. 

Then  note  the  difference  of 
height  between  the  desired  point 
and  one  of  the  given  points,  .and  ex- 
tend the  dividers  between  the  cor- 
responding numbers  on  the  scale. 
This  opening  will  be  the  distance 
to  be  set  off  on  the  plat  from  the  given  point  to  the  desired 
point. 


(148.)  Ridges  and  Thalwegs.  The  general  character  of  the 
surface  of  a  country  is  given  by  two  sets  of  lines :  the  ridge- 
lines,  or  water-shed  lines  j  and  the  "  ihalwegs"  or  lowest  lines 
of  valleys. 

The  former  are  lines  which  divide  the  water  falling  upon 
them,  and  from  which  it  passes  off  on  contrary  sides.  They 
are  the  lines  of  least  slope  when  looking  along  them  from 
above  downward ;  and  they  are  the  lines  of  greatest  slope 
when  looking  from  below  upward.  They  can  therefore  be 
readily  determined  by  the  slope-level,  etc.  They  are  the 
lines  of  least  zenith  distances  when  viewed  from  either  direc- 
tion. 

On  these  lines  are  found  all  the  projecting  or  protruding 
bends  of  the  contour-lines,  convex  toward  the  lower  ground, 
as  shown  in  Fig.  104. 

The  second  set  of  lines,  or  the  "  thalwegs,"  are  the  con- 
verse of  the  former.  They  are  indicated  by  the  water-courses 
which  follow  them  or  occupy  them.  They  are  the  lines  of 
greatest  slope  when  looked  at  from  above,  and  of  least  slope 
when  looked  at  from  below.  They  are  the  lines  of  greatest 
zenith  distance  when  viewed  from  either  direction. 


90         LEVELLING,  TOPOGRAPHY,  AXD   HIGHER  SURVEYING. 


On  these  lines  are  the  receding  or  reentering  points  of  the 
eon  tour-curves,  concave  toward  the  lower  ground. 


FIG.  104. 


The  general  system  of  the  surface  of  a  country  is  most 
easily  characterized  by  putting  down  these  two  sets  of  lines, 
and  marking  the  changes  of  slope,  especially  the  beginning 
and  the  end. 

The  most  important  points  to  be  determined  are : 

1.  At  the  top  and  bottom  of  slopes. 

2.  At  the  changes  of  slopes  in  degree. 

3.  On  the  water-shed  lines,  and  on  the  thalwegs. 

4.  On  "  cols,"  or  culminating  points  of  passes. 


FIG.  105. 


(149.)  Forms  of  Ground.  It  will  be  found  on  the  inspection 
of  a  "contour-map"  (which  shows 
ground  much  more  plainly  to  the 
eye  than  does  the  ground  itself),  that 
its  infinite  variety  of  form  may,  for 
the  purposes  of  the  engineer,  be  re- 
duced to  five :  1.  Sloping  down  on 
all  "sides;  i.  e.,  a  hill,  Fig.  105. 


FIG.  106. 


FIG.  107. 


FIRST  SYSTEM. 


91 


2.  Sloping  up  on  all  sides ;  i.  e.,  a  hollow,  Fig.  106. 

3.  Sloping  down  on  three  sides  and  up  on  one ;  i.  e.,  a 
croupe,  or  shoulder,  or  promontory,  the  end  of  a  ridge   or 
water-shed  line,  Fig.  107. 

4.  Sloping  up  on  three  sides  and  down  on  one ;  i.  e.,  a 
valley,  or  "  thalweg,"  Fig.  108. 

5.  Sloping  up  on  two  sides  and  sloping  down  on  two,  al- 
ternately;  i.  e.,  a  "pass,"  or  "col,"  or  "saddle,"  Fig.  109. 


FIG.  108. 


[X-OTE. — The  arrows  in  the  figures   indicate   the  direction  in  which  water 
would  run.] 

(150.)  Sketching  Ground  by  Contours.  A  valuable  guide  is, 
the  observation  that  the  lines  are  perpendicular  to  the  water- 
shed lines  and  thalwegs.  Note  especially  the  contour-lines  at 
the  bottoms  of  hills  and  ridges,  and  at  the  tops  of  hollows  and 
valleys,  putting  them  down,  in  their  true  relative  positions  and 
distances,  to  an  estimated  scale. 

On  a  long  slope  or  hill,  draw  first  the  bottom  contour-line, 
and  the  top  one ;  and  then  the  middle  one ;  and  afterward 
interpolate  others.  Remember  that  two  of  them  can  never 
meet,  except  on  a  perpendicular  face ;  and  that,  if  one  of  them 
passes  entirely  around  a  hill  or  hollow,  it  will  come  back  to 
its  starting-point.  Hold  the  field-book  so  that  the  lines  on  it 
have  their  true  direction. 


(151.)   Ambiguity.     In    contour-maps   of   ground,   if   the 
heights  of  the  contour-lines  are  not  written  upon  them,  it  may 


92          LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

be   doubtful  which  are  the  highest  and  lowest;  which  are 
ridges  and  which  valleys,  etc. 

1.  Numbers  remove  this. 

2.  The  water-courses  show  the  slopes.     If'  there  are  none, 
put  some  in,  in  the  thalwegs  of  a  rough  sketch. 

3.  Put  hatchings  on  the  lower  sides  of  the  contour-lines,  as 
if  water  were  draining  off. 

4.  Tint  the  valleys  and  low  places. 

(152.)  Conventionalities.  Sometimes  the  spaces  between 
contour-lines  are  colored  with  tints  of  Indian-ink,  sepia,  etc., 
increasing  in  darkness  as  the  depth  increases. 

Ground  under  water  is  commonly  so  represented,  begin- 
ning at  the  low-water  line  and  covering  the  space  to  the  six- 
feet-deep  contour-line  with  a  dark  shade  of  Indian-ink ;  then  a 
lighter  shade  from  6  to  12;  a  still  lighter  from  12  to  18,  and 
the  lightest  from  18  to  24. 

Greater  depths  are  noted  in  fathoms  and  fractions. 

(153,)  Applications  of  Contour-Lines.  They  have  many  im- 
portant uses  besides  their  representation  of  ground : 

1.  To  obtain  vertical  sections ;  i.  e.,  profiles. 

2.  To  obtain  oblique  sections. 

3.  To  locate  roads. 

4.  To  calculate  excavation   and  embankment.     Consider 
the  contour-lines  to  represent  sections  of  the  mass  by  horizon- 
tal planes.     Then  each  slice  between  them  will  have  its  con- 
tents equal,  approximately,  to  half  the  sum  of  its  upper  and 
lower  surfaces  multiplied  by  the  vertical  distance  apart  of  the 
sections.     The  areas  may  be  obtained  as  in  L.  S.  (74)  and 
(124).     This  method  is  used  to  get  the  cubic  contents  of  a  hill 
to  be  cut  away ;  of  a  hollow  to  be  filled  up ;  of  a  great  reser- 
voir in  a  valley,  either  only  projected,  or  full  of  water,  etc. 

(154.)  Sections  by  Oblique  Planes.  This  method  was  much 
used  by  the  old  military  topographers.  It  is  picturesque,  but 
not  precise.  The  cutting-planes  are  parallel,  and  may  make 
an}r  angle  with  the  horizon. 


SECOND   SYSTEM, 


93 


CHAPTER  II. 

SECOND    SYSTEM. 

BY  LINES  OP  GREATEST  SLOPE. 

(155.)  Their  Direction.  It  is  that  which  water  would  take 
in  running  down  a  slope.  They  are  drawn  perpendicularly  to 
the  contour-lines,  and  are  the  u  lines  of  greatest  slope."  They 
are  called  "  hatchings." 

FIG.  110. 


Fig.  110  represents  an  oval  hill  by  this  system. 

(156.)  Sketching  Ground  by  this  System.  This  is  rapid  and 
effective,  but  not  precise.  In  doing  this,  hold  the  book  to 
correspond  with  your  position  on  the  ground,  and  always  draw 
toward  you.  If  at  the  top  of  a  hill,  begin  by  drawing  lines 
from  the  bottom,  and  vice  versa.  The  hatchings  are  guided 
by  contour-lines  lightly  sketched  in. 

(157.)  Details  of  Hatchings.  They  must  be  drawn  very  truly 
perpendicular  to  the  contour-lines.  But  if  the  contour-lines 
are  not  parallel,  the  hatchings  must  curve.  "When  the  con- 
tours are  very  far  apart,  as  on  nearly  level  ground,  then  pen- 
cil in  intermediate  ones. 

Hatchings  in  adjoining  rows  should  not  be  continuous,  but 
"  break  joints,"  to  indicate  the  places  of  the  contour-lines, 
which  are  usually  pencilled  in  to  guide  the  hatchings,  and 
then  rubbed  out.  The  rows  of  hatchings  must  neither  overlap 


91         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

nor  separate,  and  the  lines  should  be  made  slightly  tremulous. 
When  they  are  put  in  without  contour-lines  to  guide  them, 
take  care  never  to  let  two  rows  run  into  one ;  for  the  breaks 
between  the  rows  represent  contour-lines,  and  two  contour- 
lines  of  different  heights  can  never  meet  except  on  a  vertical 
surface. 


FIG.  111. 


CHAPTER  III. 
THIRD    SYSTEM. 

BY     SHADES     FKOM     VERTICAL     LIGHT. 

(158.)  Degree  of  Shade.  The  steeper  the  slope  is,  the  less 
light  it  receives,  in  the  inverse  ratio  of  its  length ;  i.  e.,  in- 
versely as  the  secant  of  the  angle  a  which 
it  makes  with  the  horizon,  or  directly  as 
cos.  a.  Then  the  ratio  of  the  black  to  the 
white  is,  :  :  1  —  cos.  a  :  cos.  a. 

In  practice,  the  difference  of  shade  is 
much  exaggerated. 

Tables  have  been  prepared  by  various 
nations,  establishing  the  ratio  of  black  and 
III  white. 

The  proper  degree  of  shade  may  be 
given  to  the  hills  and  hollows  on  the  map  by  various  means. 

(159.)  Shades  by  Tints.  Indian-ink,  or  sepia,  is  used.  The 
shades  are  put  on  with  proper  darkness,  according  to  a  pre- 
viously-prepared "diapason  of  tints."  The  tints  are  made 
light  for  gentle  slopes,  and  dark  for  steep  slopes,  in  a  constant 
ratio,  a  slope  of  60°  being  quite  black,  one  of  30°  a  tint 
midway  between  that  and  white,  and  so  on.  The  edges  at 
the  top  and  bottom  are  softened  off  with  a  clean  brush.  This 


THIRD  SYSTEM.  95 

is  rapid  and  effective,  but  not  very  definite  or  precise,  except 
in  combination  with  contour-lines. 

(160.)  Shades  by  Contour-Lines.  This  is  done  by  making 
the  contour-lines  more  numerous ;  i.  e.,  interpolating  new  ones 
between  those  first  determined.  One  objection  to  this  is  con- 
fusion of  these  lines  with  roads. 

(161.)  Shades  by  Lines  of  Greatest  Slope.  The  lines  of  steep- 
est slope,  i.  e.,  the  hatchings  between  the  contours,  have  their 
thickness  and  distance  apart  made  proportional  to  the  steep- 
ness of  the  slope,  in  some  definite  ratio.  This  is  the  most 
usual  method. 

The  tints  may  be  produced  by  varying  the  thickness  of  the 
hatchings,  or  their  distance  apart.  Both  are  usually  combined. 

(162.)  The  French  Method.  In  this  the  degree  of  inclination 
is  indicated  by  varying  the  distances  between  the  centres  of 
the  hatchings.  Tbe  rule  is :  the  distance  between  the  centres 
of  the  lines  shall  equal  yf^  of  an  inch,  plus  J  of  the  denomi- 
nator of  the  fraction  denoting  the  declivity  (i.  e.,  tangent  of 
the  angle  made  l>y  the  surface  of  the  ground  with  the  plane  of 
reference)  expressed  in  Jiundredths  of  an  inch. 

The  lines  are  made  heavier  as  the  slope  is  steeper,  being 
fine  for  the  most  gentle  slopes,  and  increasing  in  breadth  till 
the  blank  space  between  them  equals  J  the  breadth  of  the 
lines. 

Only  slopes  of  from  T  to  -fa  inclusive  are  represented  by 
this  method. 

(163.)  The  German,  ar  Lehmann's  Method.  He  uses  nine 
grades  for  slopes  from  0°  to  45°,  the  first  being  white  and  the 
last  black.  For  the  intermediate  slopes,  he  makes  the  white 
to  the  black  in  the  following  proportion  : 

TJie  white  :   the  black  ::  45° —  angle  of  slope  \  angle  of  slope. 

For  example,  for  30° : 

light  :  shade  ::  45°—  30°  :  30°  ::  1  :  2. 

Hence,  the  space  between  the  strokes  is  to  their  thickness, 
as  45°  minus  the  angle  of  the  slope  is  to  the  angle  of  the  slope. 

•A 


96         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

Slopes  steeper  than  45°  are  represented  by  short,  heavy  lines, 


FIG.  112. 


parallel  to  the  contour-lines,  as  shown  in  the  upper  right-hand 
corner  of  Fig.  113 — a  hill  drawn  by  Lehmann's  Method. 


FIG.  113. 


FOURTH  SYSTEM. 
(164.)  Another  Diapason  of  Tints: 


97 


Slope. 

2f 

5° 

10° 

15° 

25° 

35° 

45" 

60° 

75° 

Black. 

1 

o 

3 

* 

5 

6 

7 

8 

9 

White. 

10 

9 

8 

7. 

6 

5 

4 

3 

2 

This  distinguishes  gentle  slopes  better.  It  makes  them 
darker,  and  the  steeper  slopes  lighter,  and  provides  for  slopes 
beyond  45°.  To  use  this  standard,  make  it  on  the  edge  of  a 
strip  of  paper,  and  apply  that  to  the  map  in  various  parts, 
and  draw  a  few  lines  corresponding  to  the  slope  of  those  parts ; 
then  fill  up  the  intervening  portions  with  suitable  gradations. 
The  angle  of  the  slope  is  known  from  the  map,  since  its  tan- 
gent equals  the  vertical  distance  between  the  contours,  divided 
by  the  horizontal  distance.  A  scale  can  be  made  for  any 
given  vertical  distance. 


FOUR  IE    SYSTEM. 

__,    y 
BY     SHADES     PRODUCED     BY     OBLIQUE     LIGHT. 

(165.)  Light  is  supposed  to  fall  from  the  upper  left-hand 
corner,  as  in  drawing  an  "  elevation,"  although  the  map  is  in 
plan.  Then  slopes  facing  the  light  will  have  a  light  tint,  and 
those  on  the  opposite  side  a  dark  tint. 

This  is  picturesque,  but  not  precise.  It  gives  apparent 
"relief"  to  the  ground,  but  does  not  show  the  degree  of 
steepness. 

The  shades  may  be  produced,  as  in  the  last  method,  by 
any  means — tints,  contours,  or  hatchings. 
,     By  making  a  map  with  contour-lines,  and  shaded  obliquely, 
it  will  be  both  effective  and  precise. 
7 


98         LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 


CHAPTEE  IY. 

CONVENTIONAL     SIGNS. 

(166.)  Signs  for  Natural  Surface.  Sand  is  represented  by 
fine  dots  made  with  the  point  of  the  pen  ;  gravel,  by  coarser 
dots.  Rocks  are  drawn  in  their  proper  places,  in  irregular 
angular  forms,  imitating  their  true  appearance  as  seen  from 
above.  The  nature  of  the  rocks,  or  the  geology  of  the  country, 
may  be  shown  by  applying  the  proper  colors,  as  agreed  on  by 
geologists,  to  the  back  of  the  map,  so  that  they  may  be  seen 
by  holding  it  up  against  the  light,  while  they  will  thus  not 
confuse  the  usual  details. 

(167.)  Signs  for  Vegetation.  Woods  are  represented  by  scol- 
loped circles,  irregularly  disposed,  imitating  trees  seen  "in 

plan,"  and  closer  or  farther  apart 
according  to  the  thickness  of  the 
forest.  It  is  tfsual  to  shade  their 
lower  and  right-hand  sides,  and  to 
Li  represent  their  shadows,  as  in  the 
figure,  though,  in  strictness,  this  is 
inconsistent  with  the  hypothesis  of 
vertical  light,  usually  adopted  for  "  hill-drawing."  For  pine 
and  similar  forests,  the  signs  may  have  a  star-like  form,  as  on 
the  right-hand  side  of  the  figure.  Trees  are  sometimes  drawn 
"in  elevation,"  or  sideways,  as  usually  seen.  This  makes 
them  more  easily  recognized,  but  is  in  utter  violation  of  the 
principles  of  mapping  in  horizontal  projection,  though  it  may 
be  defended  as  a  pure  convention.  Orchards  are  represented 
by  trees  arranged  in  rows.  Bushes  may  be  drawn  like  trees, 
but  smaller. 


FIG.  115. 


jlllt.  ^ 


slUlUr'  ->w«-^,,w  ***^\/i».  ^ik-" 


VV  V 

vvv 
\  vv 

W!/ 
\W 


CONVENTIONAL  SIGNS.  99 

Grass-land  is  drawn  with  irregularly  scattered  groups  of 
short  lines,  as  in  the  figure,  the  lines  be- 
ing arranged  in  odd   numbers,  and  so 
that  the  top  of  each  group  is  convex, 
and  its  bottom  horizontal  or  parallel  to 
the  base  of  the  drawing.     ^Meadows  are 
sometimes  represented  by  pairs  of  di- 
verging lines  (as  on  the  right  of  the  fig- 
ure), which  may  be  regarded  as  tall  blades  of  grass     Unculti- 
vated land  is  indicated  by  appropriately  intermingling  the 
signs  for  grass-land,  bushes,  sand,  and  rocks. 
Cultivated  land  is  shown  by  parallel  rows  of 
broken  and   dotted  lines,  as  in  the  figure, 
representing  furrows.     Crops  are  so  tempo- 
rary that  signs    for  them  are  unnecessary,  i^---—---™-------" 

though  often  used.     They  are  usually  imita-  ^^'2 

tive,  as  for  cotton,  sugar,  tobacco,  rice,  vines,  ~~~    -------"•; 

hops,  etc.     Gardens  are  drawn  with  circular  and  other  beds 
and  walks. 

(168.)  Signs  for  Water.     The  /Sea-coast  is   represented  by 
Irawing  a  line  parallel  to  the  shore,  following  all  its  windings 

FIG.  117. 


FIG.  116. 


and  indentations,  and  as  close  to  it  as  possible ;  then  another 
parallel  line  a  little  more  distant ;  then  a  third  still  more  dis- 


100      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


Fio.  118. 


FIG.  119. 


tant,  and  so  on,  as  in  Fig.  117.  If  these  lines  are  drawn  from 
the  low-tide  mark,  a  similar  set  may  be  drawn  between  that 
and  the  high-tide  mark,  and  dots,  for  sand,  be  made  over  the 
included  space. 

Rivers  have  each  shore  treated  like  the 
sea-shore,  as  in  Fig.  118. 

Brooks  would  be  shown  by  only  two 
lines,  or  one,  according  to  their  magnitude. 
Ponds  may  be  drawn 
like  sea-shores,  or  rep- 
resented by  parallel 
horizontal  lines  ruled 
across  them.  Marshes 
and  Swamps  are  repre- 
sented by  an  irregular 
intermingling  of  the  preceding  sign  with 
that  for  grass  and  bushes,  as  shown  in  Fig.  119. 

(169.)  Colored  Topography.  The  conventional  signs  which 
have  been  described,  as  made  with  the  pen,  require  much  time 
and  labor.  Colors  are  generally  used  by  the  French  as  sub- 
stitutes for  them,  and  combine  the  advantages  of  great  rapidity 
and  effectiveness.  Only  three  colors  (besides  Indian-ink)  are* 
required,  viz.:  Gamboge  (yellow),  Indigo  (blue),  and  Lake 
(scarlet),  Sepia,  Burnt  Sienna,  Yellow  Ochre,  Eed  Lead,  and 
Vermilion,  are  also  sometimes  used.  The  last  three  are  diffi- 
cult to  work  with.  To  use  these  paints,  moisten  the  end  of  a 
cake  and  rub  it  up  with  a  drop  of  water,  afterward  diluting 
this  to  the  proper  tint,  which  should  always  be  light  and  deli- 
cate. To  cover  any  surface  with  a  uniform  flat  tint,  use  a 
large  earners-hair  or  sable  brush,  keep  it"  always  moderately 
full,  incline  the  board  toward  you,  previously  moisten  the 
paper  with  clean  water  if  the  outline  is  very  irregular,  begin 
at  the  top  of  the  surface,  apply  a  tint  across  the  upper  part, 
and  continue  it  downward,  never  letting  the  edge  dry.  This 
last  is  the  secret  of  a  smooth  tint.  It  requires  rapidity  in 


CONVENTIONAL  SIGNS.  101 

returning  to  the  beginning  of  a  tint  to  continue  it,  and  dex- 
terity in  following  the  outline.  Marbling,  or  variegation,  is 
produced  by  having  a  brush  at  each  end  of  a  stick,  one  for 
each  color,  and  applying  first  one,  and  then  the  other,  beside 
it  before  it  dries,  so  that  they  may  blend,  but  not  mix,  and 
produce  an  irregularly-clouded  appearance.  Scratched  parts 
of  the  paper  may  be  painted  over  by  first  applying  strong 
alum-water  to  the  place. 

The  conventions  for  Colored  Topography,  adopted  by  the 
French  military  engineers,  are  as  follows:  WOODS,  yellow  • 
using  gamboge  and  a  very  little  indigo.  GRASS-LAND,  green  ; 
made  of  gamboge  and  indigo.  CULTIVATED  LAND,  brown  •  lake, 
gamboge,  and  a  little  Indian-ink;  "burnt  sienna"  will  an- 
swer. Adjoining  fields  should  be  slightly  varied  in  tint. 
Sometimes  furrows  are  indicated  by  strips  of  various  colors. 
GARDENS  are  represented  by  small  rectangular  patches  of 
brighter  green  and  brown.  UNCULTIVATED  LAND,  marbled  green 
and  light  brown.  BRUSH,  BRAMBLES,  etc.,  marbled  green  and 
yellow.  HEATH,  FURZE,  etc.,  marbled  green  sn&pmk.  YINE- 
YARDS,  purple  ;  lake  and  indigo.  SANDS,  a  light  brown  ;  gam- 
boge and  lake  ;  "  yellow  ochre  "  will  do.  LAKES  and  RIVERS, 
light  blue,  with  a  darker  tint  on  their  upper  and  left-hand 
sides.  SEAS,  dark  blue,  with  a  little  yellow  added.  MARSHES, 
the  blue  of  water,  with  spots  of  .grass,  green,  the  touches  all 
lying  horizontally.  ROADS,  brown; between  the  tints  for 
sand  and  cultivated  ground,  with  more  Indian-ink.  HILLS, 
greenish  brown  ;  gamboge,  indigo,  lake,  and  Indian-ink.  WOODS 
may  be  finished  up  by  drawing  the  trees  as  in  Art.  (167),  and 
coloring  them  green,  with  touches  of  gamboge  toward  the 
light  (the  upper  and  left-hand  side),  and  of  indigo  on  the  op- 
posite side. 

(170.)  Signs  for  Miscellaneous  Objects.  Too  great  a  number 
of  these  will  cause  confusion.  A  few  leading  ones  will  be 
given : 


102      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


Signal  of  survey, 

/Li  120 

Telegraph, 

rUli1  121 

Court  house, 

Jtl 

7 

m  122 

Post  office, 

J^U 

«*/             / 

EM     123 

Tavern, 

124 

Blacksmith's  shop, 

tez%%     125 

G-uide  board, 

t       126 

Quarry, 

X      12V 

Grist  mill, 

G      128 

FIG.  138. 

Ill      !I!I 

^^3lZ     Stone  bridge, 

jll|l  i      llil| 
I.  !   '   .1^     Wooden  bridge, 

Saw  mitt, 
Wind  mill, 
Steam  mill, 
Furnace, 
Woollen  factory, 
Cotton  factory, 
G-lass  works, 
Church, 
Grave  yard, 


C||  12 


129 


130 


131 


132 


133 


134 


135 


136 


m 


Aqueduct, 


An  ordinary  house  is  drawn 
in  its  true  position  and  size, 
and  the  ridge  of  its  roof  shown, 
if  the  scale  of  the  map  is  large 
Suspension  bridge,  enough.  On  a  very  small  scale, 
a  small  shaded  rectangle  rep- 
resents it.  If  colors  are  used, 
buildings  of  masonry  are  tinted 
a  deep  crimson  (with  lake),  and 
those  of  wood  with  Indian-ink. 
Their  lower  and  right-hand  sides 
are  drawn  with  heavier  lines. 
Fences  of  stone  or  wood,  and 
hedges,  may  be  drawn  in  imita- 
tion of  the  realities  ;  arid,  if  de- 
sired, colored  appropriately. 

Mines  may  be  represented 
by   the    signs   of   the   planets, 

which  were  anciently  associated  with  the  various  metals.   The 

signs  here  given  represent  respectively  : 


Ford  for  carriages, 


Ford  for  horses. 


Gold. 

a 


Silver. 

E> 


Iron. 

$ 


Copper. 
2 


Tin. 

1C 


Lead.       Quicksilver. 


CONVENTIONAL  SIGNS.  103 

A  large  black  circle,  •  ,  may  be  used  for  Coal. 

Boundary-lines,  of  private  properties,  of  townships,  of 
counties,  and  of  StateSj  may  be  indicated  by  lines  formed  of 
various  combinations  of  short  lines,  dots,  and  crosses,  as  below.1 


(171.)  Scales.  The  scale  to  which  a  topographical  map 
should  be  drawn,  depends  on  several  considerations.  The 
principal  ones  are  these :  It  should  be  large  enough  to  express 
all  necessary  details,  and  yet  not  so  large  as  to  be  unwieldy. 
The  scale  should  be  so  chosen  that  the  dimensions  measured 
on  the  ground  can  be  easily  converted,  without  calculation, 
into  the  corresponding  dimensions  on  the  map. 

In  the  United  States  Engineer  service,  the  following  scales 
are  prescribed : 

General  plans  of  buildings,  1  inch  to  10  feet  (1  :  120). 

Maps  of  ground,  with  horizontal  curves  one  foot  apart,  1  inch  to  50  feet  (1  :  600). 

Topographical  maps,  one  mile  and  a  half  square,  2  feet  to  one  mile  (1  :  2,640). 

Do.  comprising  three  miles  square,  1  foot  to  one  mile  (1  :  5,280). 

Do.  between  four  and  eight  miles  square,  6  inches  to  one  mile  (1  :  10,560). 

Do.  comprising  nine  miles  square,  4  inches  to  one  mile  (1  :  15,840). 

Maps  not  exceeding  24  miles  square,  2  inches  to  one  mile  (1  :  31,680). 

Maps  comprising  50  miles  square,  1  inch  to  one  mile  (1  :  63,360). 

Maps  comprising  100  miles  square,  %  inch  to  one  mile  (1  :.  126,720). 

Surveys  of  roads,  canals,  etc.,  1  inch  to  50  feet(l  :  600). 

On  the  admirable  United  States  Coast  Survey,  all  the 
scales  are  expressed  fractionally  and  decimally.  The  surveys 
are  generally  platted  originally  on  a  scale  of  one  to  ten  or 
twenty  thousand,  but  in  some  instances  the  scale  is  larger  or 
smaller. 

1  Very  minute  directions  for  the  execution  of  the  details  of  topographical  map- 
ping, are  given  in  Lieutenant  K.  S.  Smith's  "  Topographical  Drawing."  Wiley, 
New  York. 


104:      LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

These  original  surveys  are  reduced  for  engraving  and  pub- 
lication, and,  when  issued,  are  embraced  in  three  general 
classes  :  1°,  Small  Harbor  Charts  ;  2°,  Charts  of  Bays,  Sounds ; 
and  3°,  of  the  (Jpast  General  Charts. 

The  scales  of  the  first  class  vary  from  1 : 10,000  to  1 :  60,000, 
according  to  the  nature  of  the  harbor  and  the  different  objects 
to  be  represented. 

Where  there*  are  many  shoals,  rocks,  or  other  objects,  as  in 
Nantucket  Harbor  and  Hell-Gate,  or  where  the  importance 
of  the  harbor  makes  it  necessary,  a  larger  scale  of  1  :  5,000, 
1  :  10,000,  and  1  :  20,000,  is  used.  But  where,  from  the  size 
of  the  harbor,  or  its  ease  of  access,  a  smaller  one  will 
point  out  every  danger  with  sufficient  exactness,  the  scales  of 
1  :  40,000  and  1  :  60,000  are  used,  as  in  the  case  of  New-Bed- 
ford Harbor,  Cat,  and  Ship  Island  Harbor,  New  Haven,  etc. 

The  scale  of  the  second  class,  in  consequence  of  the  large 
areas  to  be  represented,  is  usually  fixed  at  1  :  80,000,  as  in  the 
case  of  New- York  Bay,  Delaware  Bay  and  River.  Preliminary 
charts,  however,  are  issued,  of  various  scales  from  1  :  80,000 
to  1  :  200,000. 

Of  the  third  class,  the  scale  is  fixed  at  1  :  400,000  for  the 
general  chart  of  the  coast  from  Gay  Head  to  Cape  Henlopen, 
although  considerations  of  the  proximity  and  importance  of 
points  on  the  coast  may  change  the  scales  of  charts  of  other 
portions  of  our  extended  coast. 


PART  V. 
UNDEEGEOUND   OE  MINING  SURVEYING. 


(172.)  It  has  three  objects : 

1.  To  determine  the  directions  and  extent  of -the  present 
workings  of  a  mine. 

2.  To  find  a  point  on  the  surface  of  the  ground  from  which 
to  sink  a  shaft,  to  meet  a  desired  spot  of  the  underground 
workings. 

3.  To  direct  the  underground  workings  to  meet  a  shaft  or 
any  other  desired  point. 

It  attains  these  objects  by  a  combination  of  surveying  and 
levelling. 


CHAPTER  I. 

SURVEYING     AND     LEVELLING     OLD     LINES. 

(173.)  First  Object.     To  determine  the  direction  and  extent 
of  the  present  workings  of  a  mine. 
We  have  to  measure : 

1.  Azimuths,  or  directions  right  and  left. 

2.  Lengths  or  distances. 

3.  Heights,  or  distances  up  and  down,  either  by  perpen- 
dicular or  angular  levelling ;  usually,  the  latter. 


106       LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

This  being  done,  the  relative  positions  of  all  the  points  are 
known  by  their  three  rectangular  coordinates. 

They  are  referred,  1st,  to  a  vertical  plane  (which  may  be 
either  north  and  south,  or  pass  through  the  first  line  of  the 
survey) ;  2d,  to  another  vertical  plane,  perpendicular  to  the 
preceding  one  ;  and  3d,  to  a  horizontal  datum-plane. 

(174.)  The  Old  Method,  In  the  old  method  of  Mining  Sur- 
veying, a  compass 1  is  used  for  determining  the  azimuths.  One 
form  of  the  compass  used  for  this  purpose,  and  called  a  "  dial," 
is  divided  from  0°  to  360°.  In  a  "  right-hand  dial,"  so  called, 
the  numbers  run  as  on  a  watch-face ;  the  90°  point  being  then 
on  the  east.  In  a  "  left-hand  dial "  they  run  in  a  contrary 
direction.  In  each,  the  zero-point  goes  ahead,  and  the  180°- 
point  is  at  the  eye. 

The  bearings  are  taken  in  the  usual  manner,  a  lamp  being 
the  object  sighted  to,  instead  of  the  rod  used  in  work  on  the 
surface.  , 

To  test  the  accuracy  of  the  bearing  of  a  line  taken  at  one 
end  of  it,  set  up  the  compass  at  the  other  end,  or  point  sighted 
to,  and  look  back  to  a  lamp  held  at  the  first  station,  or  point 
where  the  compass  had  been  placed  originally.  The  reading 
of  the  needle  should  now  be  the  same  as  before. 

If  the  position  of  the  sights  had  been  reversed,  the  reading 
would  be  ihe'JReverse  Bearing  ;  a  former  bearing  of  N.  30°  E. 
would  then  be  S.  30°  W.,  and  so.  on. 

If  the  back-sight  does  not  agree  with  the  first  or  forward 
sight,  this  latter  must  be  taken  over  again.  If  the  same  dif- 
ference is  again  found,  this  shows  that  there  is  local  attraction 
at  one  of  the  stations  ;  i.  e.,  some  influence,  such  as  a  mass  of 
iron-ore,  ferruginous  rocks,  etc.,  which  attracts  the  needle,  and 
makes  it  deviate  from  its  usual  direction. 

.  To  discover  at  which  station  the  attraction  exists,  set  the 
compass  at  several  intermediate  points  in  the  line  which  joins 

1  For  a  complete  description  of  the  compass,  and  method  of  using  it,  the  va- 
riation of  the  magnetic  needle,  and  methods  of  determining  the  true  meridian,  see 
L.  S.  Part  III. 


SURVEYING  AND   LEVELLING   OLD   LINES. 


107 


FIG.  140. 


the  two  stations,  and  take  the  bearing  of  the  line  at  each  of 
these  points.  The  agreement  of  several  of  these  bearings, 
taken  at  distant  points,  will  prove  their  correctness. 

When  the  difference  occurs  in  a  series  of  lines,  proceed 
thus :  Let  C  be  the  station  FlG  139 

at  which  the  back-sight  to  ° 

B  differs  from  the  fore- 
sight from  B  to  C.  Since  A 
the  back-sight  from  B  to  A 
is  supposed  to  have  agreed  with  the  fore-sight  from  A  to  B, 
the  local  attraction  must  be  at  C,  and  the  forward  bearing 
must  be  corrected  by  the  difference  just  found  between  the 
fore  and  back  sights,  adding  or  subtracting  it,  according  to 
circumstances.  An  easy  method  is  to  draw  a  figure  for  the 
case,  as  in  Fig.  140.  In  it,  suppose 
the  true  bearing  of  B  C,  as  given 
by  a  fore-sight  from  B  to  C,  to  be 
1ST.  40°  E.,  but  that  there  is  local 
attraction  at  C,  so  that  the  needle 
is  drawn  aside  10°,  and  points  in 
the  direction  S'JST',  instead  of  SK 
The  back-sight  from  C  to  B  will 
then  give  a  bearing  of  N.  50°  E. ;  a 
difference,  or  correction  for  the  next 
fore-sight,  of  10°.  If  the  next  fore- 
sight, from  C  to  D,  be  K  TO0  E.,  this  10°  must  be  subtracted 
from  it,  making  the  true  fore-sight  N.  60°  E. 

A  general  rule  may  also  be  given.  When  the  "back-sight  is 
greater  than  the  fore-sight,  as  in  this  case,  subtract  the  differ- 
ence from  the  next  fore-sight,  if  that  course  and  the  preceding 
one  have  both  their  letters  the  same  (as  in  this  case,  both  be- 
ing N".  and  E.),  or  both  their  letters  different ;  or  add  the  dif- 
ference if  either  the  first  or  last  letters  of  the  two  courses  are 
different.  When  the  hack-sight  is  less  than  the  fore-sight,  add 
the  difference  in  the  case  in  which  it  has  just  been  directed  to 
subtract  it,  and  subtract  it  where  it  was  before  directed  to 
add  it. 


108      LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

When  the  compass  indicates  much  local  attraction,  the 
difference  between  the  directions  of  two  meeting  lines  (or  the 
"  angle  of  deflection  "  of  one  from  the  other),  can  still  be  cor- 
rectly measured,  by  taking  the  difference  of  the  bearings  of . 
the  two  lines,  as  observed  at  the  same  point.  For,  the  error 
caused  by  the  local  attraction,  whatever  it  may  be,  affects  both 
bearings  equally,  inasmuch  as  a  "bearing"  is  the  angle  which 
a  line  makes  with  the  direction  of  the  needle,  and  that  here 
remains  fixed  in  some  one  direction,  no  matter  what,  during 
the  taking  of  the  two  bearings.  Thus,  in  Fig.  140,  let  the  true 
bearing  of  B  C,  i.  e.,  the  angle  which  it  makes  with  the  line 
SN",  be,  as  before,  K  40°  E.,  and  that  of  CD  K  60°  E. 
The  true  "  angle  of  deflection  "  of  these  lines,  or  the  angle 
B'C  D,  is  therefore  20°.  Now,  if  local  attraction  at  C  causes 
the  needle  to  point  in  the  direction  S'N',  10°  to  the  left  of  its 
proper  direction,  BC  will  bear  K  50°  E.,  and  CD  K  70° 
E.,  and  the  difference  of  these  bearings,  i.  e.,  the  angle  of  de- 
flection, will  be  the  same  as  before. 

In  chaining,  the  leader  holds  a  lamp  in  the  same  hand 
with  the  end  of  the  chain,  so  as  to  be  put  in  line.  When  this 
is  done,  the  follower  drops  his  end  of  the  chain,  and  goes  on 
to  find  the  pin,  or  mark  made  by  the  leader,  before  the  leader 
leaves  it. 

A  gallery  of  a  mine  is  thus  surveyed  like  a  road. 

To  measure  angles  of  elevation  or  depression  of  the  floor 
of  the  gallery,  a  fine  string  or  wire  is  stretched  parallel  to  the 
slope,  and  to  it  a  divided  semicircle  is  attached,  and  the  angle 
noted  by  a  plumb-line  suspended  from  its  centre.  See  Fig.  71. 

(175.)  The  New  Method.  The  work  by  the  old  method  is 
very  imperfect,  owing  to  the  variation  of  the  magnetic  needle, 
the  liability  of  error  from  local  attraction,  and  the  want  of 
precision  in  reading  the  angles,  both  horizontal  and  vertical. 

A  transit,  or  theodolite,  should  be  used.  The  azimuthal 
and  vertical  angles  are  taken  at  the  same  time ;  the  former  on 
the  horizontal  graduated  circle,  and  the  latter  on  the  vertical 
circle. 


SURVEYING  AND  LEVELLING  OLD  LINES.  109 

Instead  of  measuring  the  angles  which  each  line  makes 
with  the  magnetic  meridian,  as  when  the  compass  is  used,  the 
angles  measured  are  those  which  each  line  makes  with  the 
preceding  one,  or  with  the  first  line  of  the  survey,  if  the  method 
of  traversing  be  adopted. 

The  polar  coordinates  given  by  the  transit  are  to  be  re- 
duced to  the  three  coordinate  planes,  to  obtain  the  rectangular 
coordinates. 

Yery  great  accuracy  can  be  obtained  by  using  three  tri- 
pods. One  would  be  set  at  the  first  station  and  sighted  back 
to  from  the  instrument  placed  at  the  second  station,  and  a 
forward  sight  be  then,  taken  to  the  third  tripod,  placed  at  the 
third  station.  The  instrument  would  then  be  set  on  this  third 
tripod,  a  back-sight  taken  to  the  tripod  remaining  on  the  sec- 
ond station,  and  a  fore-sight  taken  to  the  tripod  brought  from 
the  first  station  to  the  fourth  station,  to  which  the  instrument 
is  next  taken ;  and  so  on.  Two  lamps,  fitting  on  the  tripods, 
are  provided,  to  which  the  backward  and  forward  sights  are 
directed. 

Owing  to  the  irregularity  of  mines,  and  the  obstacles  to 
be  overcome,  great  difficulties  exist  in  mining  surveying.  One 
is  that  of  setting  up  the  transit.  When  it  cannot  be  set  upon 
the  tripod,  it  is  often  set  upon  sockets  which  are  fastened  to 
the  wall  or  roof  of  the  mine. 

(176.)  The  Mining  Transit.     In  this  the  telescope  is  on  one 
side,  as  shown  in  Fig.  141,  and  is  balanced  by  a  weight  on  the 
opposite  side.    The  advantage  of 
this  form  is,  that  sights  may  be 
taken  vertically  up  or  down,  as  is 
sometimes  necessary  in  connect- 
ing the  underground  surveys  with 
those  on  the  surface. 
i 

(177.)  Mapping.  The  galleries 
of  a  mine  on  the  same  "level" 
may  be  platted  in  the  same  manner  as  a  road  or  stream,  etc. 


HO       LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

When  different  "levels"  are  to  be  represented,  with  their 
connecting  shafts,  etc.,  "  isometrical  projection "  has  been 
used,  but  "military  or  cavalier  projection"  is  best. 


CHAPTEE  II. 

LOCATING     NEW     LINES. 

(178.)  Second  Object,  To  determine,  on  the  surface  of  the 
ground,  where  to  sink  a  shaft  to  meet  a  desired  point  in  the 
underground  workings. 

To  do  this,  repeat  on  the  surface  of  the  ground  the  survey 
made  under  it ;  i.  e.,  trace  on  it  the  courses  and  distances  of 
the  galleries,  or  their  equivalents.  Art.  (182). 

The  chief  difficulty  is  to  get  a  starting-point,  and  to  deter- 
mine the  direction  of  the  first  line. 

(179.)  When  the  Mine  is  entered  by  an  Adit,  Fig.  142.  Set 
the  theodolite  at  the  entrance,  and  get  the  direction  of  the  adit, 

FIG.  143. 


and  prolong  it  up  the  hill ;  i.  e.,  in  the  same  vertical  plane. 
The  third  adjustment  is  here  important.     See  Art.  (93). 

If  the  line  has  to  be  prolonged  by  setting  the  instrument 
farther  on,  the  second  adjustment  is  important.    Art.  (93). 


LOCATING  NEW  LINES. 


Get 


Fia.  143. 


(180.)  When  the  Mine  is  entered  by  a  Shaft, 
netic  bearing  of  the  first  underground  line,  at  the  bottom  of 
the  shaft,  with  great  care.  Bring  up  the  end  of  the  line 
through  the  shaft  by  a  plumb-line,  and  set  the  compass  over 
this  point.  Set  out  a  line  with  the  same  bearing  and  length 
as  the  first  underground  line,  and  repeat  the  succeeding  courses. 

WHEN  THE  COMPASS  CANNOT  BE  SET  OVEK  THE  POINT,  proceed 

thus :  1st.  Find,  by  trial,  a 
spot,  as  B  (Fig.  143),  which 
is  in  the  correct  course,  and 
measure  off  a  distance  equal 
to  the  length  of  the  first  un- 
derground course,  and  then 
proceed  as  before. 

2d.  Otherwise.  —  Set  up 
anywhere,  as  at  A',  Fig.  144, 
take  the  bearing  and  distance 

of  A  from  A' ;  run  a  line,  corresponding  with  the  one  under- 
ground, from  A!  to  B'.  Eepeat  the  course  A'  A  from  B'  B ; 
then  A  B  is  the  desired  line. 


FIG.  144. 


...B 


(181.)  To  dispense  with  the  Magnetic  Needle.  First  Method. 
Let  down  two  plumb-lines  on  opposite  sides  of  the  shaft,  so 
that  their  lower  ends  shall  be  very  precisely  in  the  under- 
ground line.  The  plumbs  may  be  immersed  in  water  to  pre- 
vent vibration.  The  plumb-lines  at  the  top  of  the  shaft  will 
give  the  required  line  on  the  surface ;  but  its  shortness  is  bad. 

Second  Method. — Set,  by  repeated  trials,  two  transits  on 

,  opposite  sides  of  the  shaft,  so  that  they  shall  at  the  same  time 

point  to  one  another,  and  each,  also,  to  one  of  two  points  in 


112       LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

the  underground  line.  They  will  then  give  the  direction  of 
the  line  above-ground. 

Third  Method. — If  the  telescope  of  the  transit  be  eccen- 
tric, as  in  Fig.  141,  set  the  instrument  on  a  platform  over  the 
mouth  of  the  shaft,  so  that  the  line  of  collimation  of  the  tele- 
scope shall  be  in  the  same  vertical  plane  with  two  points  in 
the  underground  line,  on  opposite  sides  of  the  shaft.  When 
the  instrument  is  so  placed  that,  in  turning  the  telescope,  the 
intersection  of  the  cross-hairs  strikes  the  two  points  in  the 
underground  line,  the  line  of  sight,  when  directed  along  the 
surface,  will  give  the  required  line. 

(182.)  Having  determined  the  first  line,  the  courses  of  the 
underground  survey  may  be  repeated  on  the  surface ;  or  the 
bearing  and  length  of  a  single  line  be  calculated,  which  shall 
arrive  at  the  desired  point. 

Let  the  zigzag  line,  AB,  BC,  CD,  DZ,  Fig.  145,  be  the 
courses  surveyed  underground,  A  being  an  adit, 
or  at  the  bottom  of  a  shaft,  and  Z  the  point  to 
which  it  is  desired  to  sink  a  shaft.  It  is  required 
to  find  the  direction  and  length  of  the  straight 
D  line  A  Z. 

When  the  compass  is  used,  calculate  the  lati- 
tude and  departure  of  each  of  the  courses,  A  B, 
B  C,  etc.  The  algebraic  sum  of  their  latitudes 
will  be  equal  to  A  X,  and  the  algebraic  sum  of 
their  departures  will  be  equal  to  X  Z.  Then  is 

X  7 
tan.  Z  A  X  =  v~r  5  *•  e->  the   algebraic  sum  of 

the  departures  divided  by  the  algebraic  sum  of  the  latitudes  is 
equal  to  the  tangent  of  the  bearing.  The  length  of  the  line 
A  Z  equals  the  square  root  of  the  sum  of  the  squares  of  A  X 
and  X  Z ;  or  equals  the  latitude  divided  by  the  cosine  of  the 
bearing. 

When  the  transit  is  used,  instead  of  referring  all  of  the 
lines  to  the  magnetic  meridian,  as  in  the  preceding  case,  any 
line  of  the  survey  may  now  be  taken  as  the  meridian,  as  in 
"  traversing." 


LOCATING  NEW  LINES. 


113 


In  Fig.  146  all  of  the  courses  are  referred  to  the  first  line 
of  the  survey.    As  before,  a  right-angled  tri- 

X  Z 

angle  will  be  formed.     Tan.  Z  A  X  =         ' 


FIG.  146. 


.^''X 


and  the  length  of  A  Z  =  V  A  X'  +  X  Z* ;  or 
AX-:- cos.  XAZ. 

Two  or  more  lines  may  be  substituted 
for  the  single  line  in  the  two  preceding 
cases;  the  condition  being,  that  the  alge- 
braic  sums  of  their  latitudes  and  of  their 
departures  shall  be  equal  to  those  of  the  underground  survey. 


FIG.  147. 


(183.)  Third  Object.  To  direct  the  workings  of  a  mine  to 
any  desired  point. 

This  is  the  converse  of  the  second  object.  "We  repeat  under 
the  ground  the  courses  run  above-ground ;  or  their  equivalents, 
as  in  Art.  (182). 

In  Fig.  147,  let  A  B,  B  C,  CD,  D  Y,  be  the  present  work- 
ings of  a  mine,  and  Z  the  shaft  to  which  the 
workings  are  to  be  directed. 

Find  the  latitude  and  departure  of  A  Z.  x 
Then  the  difference  between  the  algebraic 
sum  of  the  latitudes  of  the    underground 
courses  already  run,  and  the  latitude  of  A  Z, 
is  the  latitude  of  the  required  course  ;  and 
the  difference  between  the  algebraic  sum  of 
the  departures  of  the  underground  lines,  and 
the  departure  of  A  Z,  is  the  departure  of  the   A 
required  course. 

The  length  of  Y  Z  equals  the  square  root 
of  the  sum  of  the  squares  of  its  latitude  and  departure. 


(184.)  Problems.  Most  of  the  problems  which  arise  in 
Mining  Surveying  can  be  solved  by  an  application  of  the  fa- 
miliar principles  of  geometry  and  trigonometry. 

1.  Given,  the  angle  which  a  vein  makes  with  the  horizon, 

8 


LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

and  the  place  where  it  meets  the  surface,  to  find  how  deep  a 

shaft  at  D  will  be   required  to 

FIG.  148.  x 

A  n         strike  the  vein  : 

;tan.  DAC. 


jc 

2.  Given,  the  depth  of  the  shaft 
D  C,  and  the  "  dip  "  of  the  vein,  to  find  where  it  crops  out  : 

AD^DC.cot.DAC. 

3.  Given,  the  depth  of  a  shaft  when  the  vein  "-crops  out," 
and  the  "  dip  "  of  the  vein,  to  find  the  distance  from  the  bot- 
tom of  the  shaft  to  the  vein  : 

BC  =  AB.cot.  ACB. 

If  the  ground  makes  an  angle  with  the  horizon,  then  the 
problems  involve  oblique-angled  triangles  'instead  of  right- 
angled  triangles,  as  in  the  preceding  cases.  Their  solution, 
however,  is  quite  as  simple. 

In  the  more  difficult  problems,  the  measurement  of  lines  is 
required,  one  or  both  ends  of  which  are  inaccessible.  For  a 
full  investigation,  of  this  subject,  see  "  Gillespie's  Land  Sur- 
veying," Part  VII. 

^ 

X 


<*.»  k 


PART  VI. 

THE  SEXTANT,    AND    OTHER   REFLECTING 
INSTRUMENTS. 


CHAPTER  I. 

THE      INSTRUMENTS. 

(185.)  Principle.  The  angle  subtended  at  the  eye  by  lines 
passing  from  it  to  two  distant  objects,  may  be  measured  by  so 
arranging  two  mirrors  that  one  object  is  looked  at  directly, 
and  the  other  object  is  seen  by  its  image,  reflected  from  one 
mirror  to  the  second,  and  from  the  second  mirror  to  the  eye. 
If  the  first  mirror  be  moved  so  that  the  doubly-reflected  image 
of  the  second  object  be  made  to-  cover  or  coincide  with  the 
object  seen  directly,  then  is  the  desired  angle  equal  to  twice 
the  angle  which  the  mirrors  make  with  each  other. 

PROOF. — Let  two  mirrors  be  parallel.  Then  a  ray  of  light, 
striking  one  of  them,  reflected  to  the  other,  and  reflected  again 
from  that,  would  pass  oif  in  a  direction  parallel  to  its  first 
direction. 


Let  a  equal  the  angle  between  the  incident  ray  and  the 


116      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

first  mirror.  Now,  suppose  the  first  mirror  to  be  turned  n°. 
The  incident  ray  now  makes  an  angle  with  this  mirror  n° 
greater  than  before ;  it  will  therefore  pass  off,  making  an  an- 
gle with  the  mirror  n°  greater  than  before.  But  the  mirror 
itself  now  makes  an  angle  of  n°  with  its  former  direction ; 
therefore,  the  ray  will  pass  off  at  an  angle  of  a°  +  2n°  with 
the  former  surface  .of  the  mirror,  and  in  a  direction  differing 
2n°  from  its  former  direction,  and  the  direction  of  the  ray  re- 
flected from  the  second  mirror  will  therefore  differ  %n°  from 
its  former  direction. 


If,  now,  an  eye,  placed  at  E,  sees  an  object  in  the  second 
mirror,  in  the  direction  E  H,  which  has  been  reflected  from 
two  mirrors,  then  the  line  E  H  mates  an  angle  with  the  true 
direction  of  the  line  equal  to  twice  the  angle  which  the  mir- 


THE  INSTRUMENTS. 


117 


rors  make  with  one  another.  If  the  eye  also  sees  an  object, 
directly  in  the  line  E  H,  which  apparently  coincides  with  the 
reflected  image  of  the  first  object,  then  is  the  angle,  subtended 
at  the  eye  by  the  lines  passing  to  it  from  the  two  objects, 
equal  to  twice  the  angle  which  the  two  mirrors  make  with  one 
another. 


(186.)  Description  of  the  Sextant,  Fig.  150.  The  frame  is 
usually  of  brass,  constructed  so  as  to  combine  strength  with 
lightness.  The  handle,  H,  by  which  it  is  held,  is  of  wood. 
A  B  is  a  graduated  arc ;  C  D,  the  index-arm,  is  movable  about 
a  pivot  in  the  centre  of  the  graduated  arc.  M  is  a  glass,  which 
may  be  moved  over  the  vernier  to  aid  in  reading  it.  The  in- 
dex-glass, I,  is  a  small  mirror,  attached  to  the  index-arm,  so 
as  to  be  perpendicular  to  the  plane  of  the  graduated  arc.  The 


FIG.  151. 


horizon-glass,  H,  is  attached  perpendicularly  to  the  plane  of 
the  instrument,  and  parallel  to  the  index-glass  when  the  index 
is  at  zero.  The  lower  half  of  this  glass  is  silvered,  to  make  it 
a  reflector,  and  the  upper  half  is  transparent.  T  E  is  the  tel- 


118      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

escope ;  S  S  are  sets  of  colored  glasses,  used  to  moderate  the 
light  of  the  sun,  when  that  body  is  observed. 

The  sextant  has  an  arc  of  one-sixth  of  a  circle,  and  meas- 
ures angles  up  to  1.20°,  the  divisions  of  the  graduated  arc 
being  numbered  with  twice  their  real  value,  so  that  the  true 
desired  angle,  subtended  by  the  two  objects,  is  read  off  at 
once.  The  arc  is  usually  graduated  to  10'  and  subdivided  by 

a  vernier  to  10". 

% 

(187.)  The  box  or  pocket  sextant,  shown  in  Fig.  151,  has 
the  same  glasses  as  the  larger  sextant,  enclosed  in  a  circular 
box,  about  three  inches  in  diameter.  The  lower  part,  which 
answers  for  a  handle  when  in  use,  screws  off  and  is  used  for  a 
cover,  making  the  instrument  only  half  as  deep  as  it  appears 
in  the  figure. 

The  octant  has  an  arc  of  one-eighth  of  a  circumference, 
and  measures  angles  to  90°. 

(188.)  The  Reflecting  Circle.  This  is  an  instrument  con- 
structed on  the  same  principle,  and  used  for  the  'same  pur- 
poses, as  the  sextant.  In  it  the  graduated  arc  extends  to  the 
whole  circumference,  and  more  than  one  vernier  may  be  used 
by  producing  the  index-arm  to  meet  the  circumference  in  one 
.or  two  more  points. 

(189.)  Adjustments  of  the  Sextant,  1.  To  make  the  index- 
glass  perpendicular  to  the  plane  of  the  arc: 

Bring  the  index  near  the  centre  of  the  arc,  and  place  the 
eye  near  the  index-glass,  and  nearly  in*  the  plane  of  the  arc. 
See  if  the  part  of  the  arc  reflected  in  the  mirror  appears  to  be 
a  continuation  of  the  part  seen  directly.  If  so,  the  glass  is 
•  perpendicular  to  the  plane  of  the  arc.  If  not,  adjust  it  by 
the  screws  behind  it. 

2.  To  make  the  horizon-glass  perpendicular  to  the  plane  of 
the  arc : 

The  index-glass  having  been  adjusted,  sight  to  some  well- 
defined  object,  as  a  star,  and  if,  in  moving  the  index-arm, 
one  image  seems  to  separate  from  or  overlap  the  other,  then 


THE  INSTRUMENTS.  119 

the  horizon-glass  is  not  perpendicular  to  the  plane  of  the  arc. 
It  must  be  made  so  by  the  screws  attached  to  it. 

Another  method  of  testing  the  perpendicularity  of  the 
horizon-glass  is  as  follows:  Hold  the  instrument  vertically, 
and  bring  the  direct  and  reflected  images  of  a  smooth  portion 
of  the  distant  horizon  into  coincidence.  Then  turn  the  instru- 
ment until  it  makes  an  angle  with  the  vertical.  If  the  two 
images  still  coincide,  the  glasses  are  parallel;  and,  as  the 
index-glass  has  been  made  perpendicular  to  the  plane  of  the 
arc,  the  horizon-glass  is  in  adjustment. 

3.  To  make  the  line  of  collimation  of  the  telescope  parallel 
to  the  plane  of  the  arc  : 

The  line  of  collimation  of  the  telescope  is  an  imaginary 
line,  passing  through  the  optical  centre  of  the  object-lens,  and 
a  point  midway  between  the  two  parallel  wires.  These  wires 
are  made  parallel  to  the  plane  of  the  sextant  by  revolving  the 
tube  in  which  they  are  placed. 

To  see  whether  the  line  of  collimation  of  the  telescope  is 
in  adjustment,  bring  the  images  of  two  objects,  such  as  the 
sun  and  moon,  into  contact  at  the  wire  nearest  the  instrument, 
and  then,  by  moving  the  instrument,  bring  them  to  the  other 
wire.  If  the  contact  remains  perfect,  the  line  of  collimation 
is  parallel  to  the  plane  of  the  arc ;  if  it  does  not,  the  adjust- 
ment must  be  made  by  the  screws  in  the  collar  of  the  tele- 
scope. 

4.  To  see  if  the  two  mirrors  dre  parallel  when  the  index  is 
at  zero  : 

Bring  the  direct  and  reflected  images  of  a  star  into  coin- 
cidence. If  the  index  is  at  zero,  then  no  correction  is  neces- 
sary ;  if  not,  the  reading  is  the  "index-error"  and  is  positive 
or  negative,  according  as  the  index  is  to  the  right  or  left  of 
the  zero. 

The  "  index-error "  may  be  rectified  by  moving  the  hori- 
zon-glass until  the  images  do  coincide  when  the  index  is  at 
zero,  but  it  is  usually  merely  noted,  and  used  as  a  correction, 
being  added  to  each  reading  if  the  error  is  positive,  or  sub- 
tracted from  each  reading  if  the  error  is  negative. 


120      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

(190.)  How  to  observe.  Hold  the  instrument  'so  that  its 
plane  is  in  the  plane  of  the  two  objects  to  be  observed,  and 
hold  it  loosely.  Look  through  the  eye-hole,  or  plain  tube,  or 
telescope,  at  the  left  hand  or  lower  object,  by  direct  vision, 
through  the  unsilvered  part  of  the  horizon-glass.  Then  move 
the  index-arm  till  the  other  object  is  seen  in  the  silvered  part 
of  the  horizon-glass,  and  the  two  are  brought  to  apparently 
coincide.  Then  the  reading  of  the  vernier  is  the  angle  desired. 

If  one  object  be  brighter  than  the  other,  look  at  the  former 
by  reflection.  If  the  brighter  objects  be  to  the  left  or  below, 
hold  the  instrument  upside  down. 

If  the  angular  distance  of  the  object  be  more  than  the 
range  of  the  sextant  (about  120°),  observe  from  one  of  them  to 
some  intermediate  object,  and  thence  to  the  other. 

A  good  rest  for  a  sextant  is  an  ordinary  telescope-clamp, 
through  which  is  passed  a  stick,  one  end  of  which  is  fitted 
into  a  hole  made  in  the  sextant-handle,  and  the  other  end  of 
which  is  weighted  for  a  counterpoise. 

% 

FIG,  153, 


(191.)  Parallax  of  the  Sextant,  The  angle  observed  with 
the  sextant  is  that  made  by  two  lines  :  one,  B  I,  passing  from 
the  reflected  object  to  the  index-glass,  and  which  is  thence 
reflected  to  the  horizon-glass,  and  thence  to  the  eye  ;  and  the 
other,  HE,  passing  from  the  object  directly  to  the  eye;  i.  e., 
the  angle  which  BI  produced  makes  with  H  E.  But  the  eye 
may  be  at  E'  on  either  side  of  E.  Then  we  require  the  angle 


THE  PRACTICE.  121 

which  B  E'  makes  with  H  E.  These  angles  are  the  same  only 
when  the  eye  is  in  the  same  line  with  B I  produced ;  i.  e., 
when  E7  coincided  with  E.  In  all  other  cases,  the  observed 
angle  differs  from  the  desired  angle  by  the  small  angle  E  B  E', 
which  is  called  the  parallax  of  the  instrument. 

It  is  the  angle  which  would  be  subtended  at  the  reflected 
object  B  by  the  distance  E  E'.  It  is  usually  very  small  for  dis- 
tant objects.  Thus,  at  a  mile's  distance,  1  inch  subtends  an 
angle  of  only  3  seconds,  and  of  3  minutes  at  100  feet  distance. 

To  escape  it,  if  one  object  be  distant  and  the  other  near, 
view  the  former  by  reflection.  If  both  be  near,  find  some  dis- 
tant point  in  line  with  one  of  them,  and  view  this  new  point 
by  reflection,  and  the  other  near  one  directly. 


CHAPTER  II. 

THE      PRACTICE. 

(192.)  To  set  out  Perpendiculars.  Set  the  index  at  90°. 
Hold  the  instrument  over  the  given  point  by  a  plumb-line, 
and  look  along  the  line  by  direct  vision.  Send  a  rod  in  about 
the  desired  direction,  and  when  it  is  seen  by  reflection  to  coin- 
cide with  the  point  on  the  line  looked  at  directly,  it  will  be  in 
a  line  perpendicular  to  the  given  line  at  the  desired  point. 

Conversely,  to  find  where  a  perpendicular  from  a  given 
point  would  strike  a  line : 

Set  the  index  at  90°,  and  walk  along  the  line,  looking 
directly  at  a  point  on  it,  until  the  given  point  is  seen  by  re- 
flection to  coincide  with  the  point  on  the  line.  A  plumb-line 
let  fall  from  the  eye  will  give  the  desired  point. 

(193.)  The  Optical  Square,  Fig.  153.  This  is  a  box  contain- 
ing two  mirrors,  fixed  at  an  angle  of  45°  to  each  other,  and 


122       LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

therefore  giving  an  angle  of  90°,  as  does  the  sextant  with  its 
glasses  fixed  at  that  angle.    It  is  used  to  set  out  perpendiculars. 


(194.)  To  measure  a  Line,  one  End  being  inaccessible, 
A  B  be  the  required  line,  and  B  the  inaccessible  point. 


Let 


FIG.  154. 


At  A  set  off  a  perpendicular,  A  0,  by  Art.  (192).  Then 
set  the  index  at  45°,  and  walk  backward  from  A  in  the  line 
C  A  prolonged,  looking  by  direct  vision  at  C,  until  you  arrive 
at  some  point,  D,  from  which  B  is  seen  by  reflection  to  coincide 
with  C.  Then  is  A  D  =  A  B. 

If  more  convenient,  after  setting  off  the  right  angle,  set 
the  index  at  63°  26',  and  then  proceed  as  before.  The  objects 
will  be  seen  to  coincide  when  at  some  point  D'.  Then 
A  D'  =  J  AB.  If  the  index  be  set  at  Tl°  34',  then  the  meas- 
ured distance  will  be  -J-  AB,  and  so  on. 


THE  PRACTICE. 


123* 


If  the  index  be  set  at  the  complements  of  the  above  angles, 
the  distance  measured  will  be,  in  the  first  case,  twice,  and  in 
the  second  case  three  times  the  desired  one. 

When  the  distance  AD  cannot  be  measured,  as  in  Fig. 
155,  fix  D  as  before.  Set  the  index  at  26°  34',  and  go  along 


the  line  to  E,  where  the  objects  are  seen  to  coincide  with  each 
other ;  then  is  A  E  twice  A  B,  and  hence  E  D  =  A  B. 


FIG.  156 


LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

(195.)  Otherwise.  At  A  set  off  an  angle,  as  CAD  (AD 
being  a  prolongation  of  AB).  Then  walk  along  the  line  A  C 
with  the  index  set  to  half  that  angle,  looking  at  A  directly, 
and  B  by  reflection,  till  you  come  to  some  point,  C,  at  which 
they  coincide.  Then  is  C  A  =  A  B. 

(196.)  To  measure  a  Line  when  both  Ends  are  inaccessible. 
Let  AB  be  the  required  line.  At  any  point,  Q,  measure 

Fio.  157. 


the  angle  A  C  B.  Set  the  sextant  to  half  that  angle,  and  walk 
back  in  the  line  B  C  prolonged  till  at  some  point,  D,  A  and  B 
are  seen  to  coincide,  as  in  last  problem ;  thus  making  A  C  = 
C  D.  Do  the  same  on  A  C  produced  to  some  point,  E.  Then 


(197.)  All  the  methods  for  overcoming  obstacles  to  measure- 
ment, determining  inaccessible  distances,  etc.  (L.  S.  Part  VII.), 
with  the  transit  or  theodolite,  can  be  executed  with  the  sextant. 

(198.)  To  measure  Heights.  Measure  the  vertical  angle  be- 
tween the  top  of  the  object  and  a  mark  at  the  height  of  the 
eye,  as  with  a  theodolite  or  transit,  and  then  calculate  the 
height  as  in  Part  II.,  Art.  (98). 

Otherwise.  Set  the  index  at  45°,  and  walk  backward  till 
the  mark  and  the  top  of  the  object  are  brought  to  coincide. 
Then  the  horizontal  distance  equals  the  height. 


THE   PRACTICE. 


125 


So,  too,  if  the  index  is  set  at  63°  26',  the  height  equals 
twice  the  distance,  and  so  on.  The  ground  is  supposed  to  be 
level. 

FIG.  158. 


When  the  Base  is  inaccessible :  Make  C  =  45°,  and  D  = 
26°  34'.  Then  CD  =  AB.  So,  too,  if  C  =  26°  34',  and 
D  =  18°  26'. 

This  may  be  used  when  a  river  flows  along  the  base  of  a 
kill  whose  height  is  desired,  or  in  any  other  like  circumstance. 

(199.)  To  observe  Altitudes  in  an  artificial  Horizon.     In  this 

FIG.  159 


method  we  measure  the  angle  subtended  at  the  eye  between 
the  object  and  its  image  reflected  from  an  artificial  horizon  of 


126      LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

mercury,  molasses,  oil,  or  water.  The  image  of  the  object  in 
the  mercury  is  looked  at  directly,  and  the  object  itself  is 
viewed  by  reflection.  The  object  observed  is  supposed  to  be 
so  distant  that  the  rays  from  it,  which  strike  respectively  the 
index-glass  and  the  artificial  horizon,  are  parallel ;  i.  e.,  S  and 
S',  Fig.  159,  are  the  same  point. 

Then  will  the  observed  angle  S  E  S"  be  double  the  re- 
quired angle  S  E  H. 

Demonstration. 

a  =  a',  a!  =  a",  and  a!'  =  a"'.     Hence  a!"  =  a. 
S  E  S"  =  a  +  a!"  =  2a  =  2  S  E  H. 

(200.)  When  the  sun  is  the  object  observed,  to  determine 
whether  it  is  his  upper  or  lower  limb  whose  altitude  has  been 
observed,  proceed  thus : 

Having  brought  two  limbs  to  touch,  push  the  index-arm 
from  you.  If  one  image  passes  over  the  other,  so  that  thft 
other  two  limbs  come  together,  then  you  had  the  lower  limb 
at  first.  If  they  separate,  you  had  the  upper  limb. 

In  the  forenoon,  with  an  inverting  telescope,  the  lower 
limbs  are  parting,  and  the  upper  limbs  are  approaching ;  and 
vice  versa  in  the  afternoon. 

FIG.  160. 


N 


(201.)  To  observe  very  small  altitudes  and  depressions  with 
the  artificial  horizon : 


THE  PRACTICE.  127 

Stretch  a  string  over  the  artificial  horizon.  Place  your 
head  so  that  you  see  the  string  cover  its  image  in  the  mercury. 
Then  the  eye  and  string  determine  a  vertical  plane. 

Then  observe,  looking  at  the  string  by  direct  vision,  and 
seeing  the  object  by  reflection,  and  you  have  the  angle  SEN", 
in  Fig.  160,  the  supplement  of  the  zenith  distance. 

Otherwise.  Fix  behind  the  horizon-glass  a  piece  of  white 
paper  with  a  small  hole  in  it,  and  with  a  black  line  on  it  per- 
pendicular to  the  plane  of  the  arc. 

Then  look  into  the  mercury,  so  as  to  see  in  it  the  image 
of  the  line.  Your  line  of  sight  is  then  vertical,  and  the  angle 
to  the  object  seen  by  reflection  is  measured  as  before. 

(202.)  To  measure  Slopes  with  the  Sextant  and  Artificial 
Horizon.  Let  AB  be  the  surface  of  the  ground,  and  AF  a 


horizontal  line.     Mark  two  points  equally  distant  from  the 
eye.     Measure,  by  the  preceding  method,  the  angles  0  and  (3' 4.  ^  _  ,^  ^ 
which  C  A  and  C  B  make  with  the  vertical  C  D.     Then  will^       ~  f 
half  the  difference  of  these  angles  equal  the  angle  which  the  ~9^ 


slope  makes  with  the  horizon.  j£  <&,  •-(fl7»  /J  }  * 

Demonstration.  Continue  the  vertical  line  C  D  to  meet 
the  horizontal  line  in  F,  and  draw  C  E  perpendicular  to  A  B.  *  $(} 
Then  the  triangles  C  D  E  and  A  D  F  are  similar;  being  right- 
angled  and  having  the  acute  angles  at  D  equal.  Consequently, 
the  angle  D  C  E  =  D  A  F,  which  is  the  angle  of  the  slope 
with  the  horizon.  But  D  C  E  =  J  (0'—  0),  hence  \  (0'—  0)  = 
the  angle  which  the  slope  of  the  ground  makes  with  the 
horizon.  .o  ;/  ^  / 


<"*—•»          - 

"^    B- 


~    & 


*      «0~ 


-  Ox" 


"      -'     ->v< 


Of}    -^  to*'  -t  s> 


128      LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

If  the  points  A  and  B  be  not  equally  distant  from  C,  but 
yet  far  apart,  this  method  will  still  give  a  very  near  approxi- 
mation, the  error,  which  is  additive,  being  -J-  («'  —  a). 

Demonstration. 


=  -/3-a+  90°, 


(203.)  Oblique  Angles.  When  the  plane  of  two  objects, 
observed  by  the  sextant,  is  very  oblique  to  the  horizon,  the 
observed  angle  will  differ  much  from  the  horizontal  angle 
which  is  its  horizontal  projection,  and  which  is  the  angle 
needed  for  platting.  The  projected  angle  may  be  larger  or 
smaller  than  the  observed  angle. 

This  difficulty  may  be  obviated  in  various  ways  : 

1.  Observe  the  angular  distance  of  each  object  from  some 
third  object,  very  far  to  the  right  or  left  of  both.     The  differ- 
ence of  these  angles  will  be  nearly  equal  the  desired  angle. 

2.  Note,  if  possible,  some  point  above  or  below  one  of  the 
objects,  and  on  the  same  level  with  the  other,  and  observe  to 
it  and  the  other  object. 

3.  Suspend  two  plumb-lines,  and  place  the  eye  so  that 
these  lines  cover  the  two  objects.     Then  observe  the  horizon- 
tal angle  between  the  plumb-lines. 

4.  For  perfect  precision,  observe  the  oblique  angle  itself, 
and  also  the  angle  of  elevation  or  depression  of  each  of  the 
objects.     With  these  data  the  oblique  angle  can  be  reduced  to 
its  horizontal  projection,  either  by  descriptive  geometry  or 
more  precisely  by  calculation,  thus  : 

Let  A  H  B  be  the  observed  angle,  and  A'  H  B'  the  required 
horizontal  angle. 

Conceive  a  vertical  H  Z,  and  a  spherical  surface,  of  which 
H,  the  vertex  of  the  angle,  is  the  centre.  Then  will  the  ver- 


THE  PRACTICE. 


129 


tical  planes,  A  H  A'  and  B  H  B',  and  the  oblique  plane  A  H  B, 
cut  this  sphere  in  arcs  of  great  circles,  Z  A",  Z  B",  and  A"  B"' 


FIG.  162. 


thus  forming  a  spherical  triangle,  A"  Z  B",  in  which  A"  B"  =  h 
measures  the  observed  angle  ;  Z  A"  =  Z  'measures  the  zenith 
distance  of  the  point  A ;  and  Z  B"  ==  Z';  measures  the  zenith 
distance  of  the  point  B. 

These  zenith  distances  are  observed  directly,  or  given  by 
the  observed  angles  of  elevation  or  depression.  Then  we  have 
the  three  sides  of  the  triangle  to  find  the  angle Z=  A'  H  B7. 

Calling  P  the  half  sum  of  the  three  sides  we  have : 


sin.  Z^  sin.  71 

An  approximate  correction,  when  the  zenith  distances  do 
not  differ  from  90°  by  more  than  2°  or'  3°,  is  this  : 


7_t_7/\«  f*7  _  7'\a 

(90°  -  ±±-  J  tang.  iA.  sin.  V  -  (—£-)   cot.  £  h.  sin.  1". 

The  quantities  in  the  parentheses  are  to  be  taken  in  sec- 
onds. 

The  answer  is  in  seconds,  and  additive. 

(204.)  The  advantages  of  the  sextant  over  the  theodolite 
are  these  : 

1  See  Jackson's  Trigonometry,  page  65,  Fifth  Case. 


130        LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

1.  It  does  not  require  a  fixed  support,  but  can  be  used 
while  the  observer  is  on  horseback,  or  on  a  surface  in  motion, 
as  at  sea. 

2.  It  can  take  simultaneous  observations  on  two  moving 
bodies,  as  the  moon  and  a  star. 

It  can  also  do  all  that  the  theodolite  can.  Its  only  defect 
is  in  observing  oblique  angles  in  some  cases.  By  these  prop- 
erties it  determines  distances,  heights,  time,  latitude,  longi- 
tude, and  true  meridian,  and  thus  is  a  portable  observatory. 


PART  VII. 

MARITIME    OR    HYDROGRAPHICAL 
SURVEYING. 


(205.)  THE  •Irject  *f  this  is  tt  fix  the  positions  »f  the  deep 
and  shallow  points  in  harbors,  rivers,  etc.,  and  thus  to  discover 
and  record  the  shoals,  rocks,  channels,  and  other  important 
features  of  the  locality. 

The  relative  positions  of  prominent  points  on  the  shore 
are  very  precisely  determined  by  "  Trigonometrical  Survey- 
ing," Part  VIII.  These  form  the  basis  of  operations,  and 
afford  the  means  of  correcting  the  results  obtained  by  the  less 
accurate  methods  employed  for  filling  in  the  details. 


CHAPTEE   I. 


THE     SHORE     LINE. 


(206.)  The  High-water  Line.  The  principal  points  on  the 
high-water  line  are  determined  by  triangulating,  Art.  (233). 
The  sections  between  these  points  are  surveyed  with  the  com- 
pass and  chain ;  by  running  a  series  of  straight  lines  so  as  to 
follow,  approximately,  the  shore  line,  and  taking  offsets  from 


132       LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

the  straight  lines  of  the  survey  to  the  bends  in  the  shore  line. 
The  straight  lines  can  be  more  accurately  determined  by 
"  traversing  "  with  the  transit.  Art.  (94). 

(207.)  The  Low  Water-Line.  In  "  tidal- waters  "  this  is  more 
difficult,  because  low  and  bare  for  only  a  short  time.  The 
survey  is  best  made  with  the  sextant,  observing  from  promi- 
nent points  to  three  signals,  by  the  trilinear  method — Art. 
(213) — and  sketching  by  the  eye  bends  of  the  shore  between 
the  stations  observed  from. 

There  should  be  one  to  observe  and  one  to  record.  Let  1 
and  2,  Fig.  163,  be  two  points  on  the  low-water  line,  whose 
position  it  is  desired  to  determine.  The  observations  taken 
will  be  as  follows  : 

(1.)     A  and  B     ...     18° 

B  and  C     .     .     .     20° 
(2.)     B  and  C     .     .     .     15° 
C  and  D     ...     45° 

When  the  shore  is  inaccessible,  a 
base  line  must  be  measured  on  the 
water,  and  points  on  the  shore  fixed  by  angles  from  its  ends, 
as  in  Art.  (232). 

(208.)  Measuring  the  Base.  1.  By  sound.  Sound  travels 
at  the  rate  of  1,090  feet  per  second,  with  the  temperature  at 
30°  Fahr.  For  higher  or  lower  temperatures,  add  or  subtract 
1^-  foot  for  each  degree.  If  the  wind  blows  with  or  against 
the  movement  of  the  sound,  its  velocity  must  be  added  or 
subtracted.  If  it  blows  obliquely,  the  correction  will  be  its 
velocity  multiplied  by  the  cosine  of  the  angle  which  the  direc- 
tion of  the  wind  makes  with  the  direction  of  the  sound. 

2.  By  measuring  with  the  sextant  the  angular  height  of 
the  mast  of  a  vessel,  then  we  have : 

Distance  =  height  of  mast  -~  tan.  of  the  angle. 

v 
I       ^>v. 

*-  = 


SOUNDINGS. 


133 


3.  By  astronomical  observations  at  points  50  miles  apart, 
more  or  less,  determining  their  latitudes  and  longitudes,  and 
hence  knowing  their  distance  and  bearings.  A  vessel  may  be 


anchored  at  various  points  between  A'  and  B',  and  thus  new 
base  lines  be  formed,  from  the  ends  of  which  to  triangulate  to 
points  on  the  shore. 


CHAPTER  II. 


SOUNDINGS. 


(209.)  IN  a  river  or  narrow  water,  the  soundings  may  be 
taken  in  zigzag  lines,  from  shore  to  shore,  at  equal  intervals 
of  time,  as  in  Fig.  165. 


Fio.  165. 


134      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

(210.)  On  a  Sea-coast.  The  position  of  the  boat  in  the 
water,  when  the  soundings  are  taken,  must  be  determined  at 
regular  intervals.  This  is  done  in  various  ways. 

(211.)  From  the  Shore.  By  observing  with  a  compass  or 
transit  to  the  boat  from  stations  on  the  shore,  at  a  given  signal 
or  fixed  time. 

FIG.  166. 


Its  place  is  then  fixed  as  in  Art.  (232).  Two  observers 
are  necessary.  Three  are  better,  as  the  third  checks  the  other 
two.  This  is  accurate  in  theory,  but  not  in  practice,  simulta- 
neous observations  being  impracticable,  and  confusion  fre- 
quent. Also,  more  men  are  required. 

(212.)  From  the  Boat  with  a  Compass.  Establish  signals 
along  the  shore,  Art.  (240),  distinguish  them  by  colors,  or  by 
the  number  of  cross-pieces  on  the  staff,  thus :  -f  ±  + ,  and 
observe  to  them  from  the  boat  with  a  prismatic  compass  (L.  S. 
232),  or  Burnier's  compass,  Art.  (96).  The  place  of  the  boat 
is  then  determined,  and  may  be  fixed  on  the  map  by  drawing, 
from  the  two  known  points,  lines  having  the  opposite  bearings, 
and  their  intersection  will  be  the  required  point.  This  is 
rapid  and  easy,  but  not  precise. 

(213.)  From  the  B'oat  with  the  Sextant.  This  is  the  trilinear 
method,  and  is  the  best.  Two  observers,  or  two  sextants  with 
one  observer,  are  necessary. 


SOUNDINGS. 


135 


(214.)  TEILINEAB  SURVEYING  is  founded  on  the  method  of 
determining  the  position  of  a  point 
by  measuring  the  angles  between  three  ^ 
lines  conceived  to  pass  from  the  re- 
quired point  to  three  known  points. 
Thus,  in  the  figure,  the  point  P  is  de- 
termined by  the  angles  A  P  B  and 
B  P  C,  the  points  A,  B,  and  C,  be- 
ing known.  To  fix  the  place  of  the 
point  from  these  data  is  known  as  the 
"problem  of  the  three  points."  It 
will  be  here  solved  geometrically,  instrument  ally,  and  analyt- 
ically. 

(215.)  Geometrical  Solution,  Let  A,  B,  and  C,  be  the  known 
objects  observed  from  S,  the  angles  A  S  B  and  B  S  C  being 
there  measured.  To  fix  this  point,  S,  on  the  plat  containing 

FIG.  168. 


A,  B,  and  C,  draw  lines  from  A  and  B,  making  angles  with 
AB,  each  equal  to  90°— A  SB.     The  intersection  of  these 
lines  at  O  will  be  the  centre  of  a  circle  passing  through  A  and 

B,  in  the  circumference  of  which  the  point  S  will  be  situated.1 

1  For,  the  arc  A  B  measures  the  angle  A  0  B  at  the  centre,  which  angle  =  180° 
—  2  (90°  —  A  S  B)  =  2  A  S  B.  Therefore,  any  angle  inscribed  in  the  circumfer- 
ence and  measured  by  the  same  arc  is  equal  to  A  S  B. 


136      LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

Describe  this  circle.  Also,  draw  lines  from  B  and  C,  making 
angles  with  B  C,  each  equal  to  90°  —  B  S  C.  Their  intersec- 
tion, O',  will  be  the  centre  of  a  circle  passing  through  B  and 
C.  The  point  S  will  lie  somewhere  in  its  circumference,  and, 
therefore,  in  its  intersection  with  the  former  circumference. 
The  point  is  thus  determined : 

In  the  figure  the  observed  angles,  A  S  B  and  B  S  C, 
are  supposed  to  have  been  respectively  40°  and  60°.  The 
angles  set  off  are  therefore  50°  and  30°.  The  central  angles 
are  consequently  80°  and  120°,  twice  the  observed  an- 
gles. 

The  dotted  lines  refer  to  the  checks  explained  in  the  latter 
part  of  this  article. 

When  one  of  the  angles  is  obtuse,  set  off  its  difference  from 
90°  on  the  opposite  side  of  the  line  joining  the  two  objects  to 
that  on  which  the  point  of  observation  lies. 

When  the  angle  ABC  is  equal  to  the  supplement  of  the 
sum  of  the  observed  angles,  the  position  of  the  point  will  be 
indeterminate,  for  the  two  centres  obtained  will  coincide,  and 
the  circle  described  from  this  common  centre  will  pass  through 
the  three  points,  and  any  point  of  the  circumference  will  fulfil 
the  conditions  of  the  problem. 

A  third  angle,  between  one  of  the  three  points  and  a  fourth 
point,  should  always  be  observed,  if  possible,  and  used  like 
the  others,  to  serve  as  a  check. 

Many  tests  of  the  correctness  of  the  position  of  the  point 
determined  may  be  employed.  The  simplest  one  is,  that  the 
centres  of  the  circles,  O  and  O',  should  lie  in  the  perpendicu- 
lars drawn  through  the  middle  points  of  the  lines  A  B  and  B  C. 

Another  is,  that  the  line  B  S  should  be  bisected  perpendic- 
ularly by  the  line  O  O'. 

A  third  check  is  obtained  by  drawing  at  A  and  C  perpen- 
diculars to  AB  and  CB,  and  producing  them  to  meet  B  O 
and  B  O',  produced  in  D  and  E.  The  line  D  E  should  pass 
through  S  ;  for,  the  angles  B  S  D  and  B  S  E  being  right  angles, 
the  lines  D  S  and  S  E  form  one  straight  line. 

The  figure  shows  these  three  cheeks  by  its  dotted  lines. 


SOUXDIXGS.  137 

(216.)  Instrumental  Solution.  The  preceding  process  is  te- 
dious where  many  stations  are  to  be  determined.  They  can 
be  more  readily  found  by  an  instrument  called  a  Station- 
pointer,  or  Chorogra/ph.  It  consists  of  three  arms,  or  straight- 
edges, turning  about  a  common  centre,  and  capable  of  being 
set  so  as  to  make  with  each  other  any  angles  desired.  This  is 
effected  by  means  of  graduated  arcs  carried  on  their  ends,  or 
.by  taking  off  with  their  points  (as  with  a  pair  of  dividers)  the 
proper  distance  from  a  scale  of  chords  constructed  to  a  radius 
of  their  length.  Being  thus  set  so  as  to  make  the  two  observed 
angles,  the  instrument  is  laid  on  a  map  containing  the  three- 
given  points,  and  is  turned  about  till  the  three  edges  pass 
through  these  points.  Then  their  centre  is  at  the  place  of  the 
station,  for  the  three  points  there  subtend  on  the  paper  the 
angles  observed  in  the  field. 

A  simple  and  useful  substitute  is  a  piece  of  transparent 
paper,  or  ground  glass,  on  which  three  lines  may  be  drawn  at 
the  proper  angles  and  moved  about  on  the  paper  as  before. 

(217.)  Analytical  Solution.  The  distances  of  the  required 
point  from  each  of  the  known  points  may  be  obtained  analyt- 
ically. Let  AB  =  0;  BC  =  a;  ABC  =  B;  ASB  =  S; 
B  S  C  =  S'.  Also,  make  T  =  360°  -  S  -  S'  -  B.  Let 
BAS  =  U;  BCS  =  Y.  Then  we  shall  have  : 


Cot.  U  =  cot.  T 


sin.  S  .  cos.  T 
Y  =  T  -  U. 

c  .  sin.  U  a  .  sin.  Y 

-  ;  -  ~  —  ?    OT%  —  —  ~»  -  cT7~-- 

sin.  S  sin.  S' 

c  .  sin.  A  B  S  a.  sin.  CBS 

fe  A  =    -   —  ~.  -  5;  --  .  b  U  —   -  -,  --  777  -  . 

sin.  S  sin.  S 

Attention  must  be  given  to  the  algebraic  signs  of  the  trig- 
onometrical functions.      -^i    -p      3c*Q 


138      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

Example.  A  S  B  =  33°  45';  B  S  C  =  22°  30';  AB  =  600 
feet;  B 0  =  400  feet;  A 0  =  800  feet.  Kequired  the  dis- 
tances and  directions  of  the  point  S  from  each  of  the  stations. 

In  the  triangle  ABC,  the  three  sides  being  known,  the 
angle  A  B  0  is  found  to  be  104°  28'  39".  The  formula  then 
gives  the  angle  B  A  S  =  U  =  105°  8'  10" ;  whence  B  0  S  is 
found  to  be  94°  8'  11" ;  and  S  B  =  1042.51 ;  S  A  =  710.193 ; 
and  S  0  =  934.291. 

(218.)  Between  Stations.  Positions  of  the  boat  are  thus  ob- 
served only  at  considerable  distances  apart,  and  the  boat  is 
rowed  from  one  of  these  points  to  a  second  -one,  and  soundings 
taken  at  regular  intervals  of  time  between  them. 


FIG. 


The  distance  apart  of  the  soundings  depends  on  the  regu- 
larity of  the  bottom,  the  depth  of  the  water,  and  the  object  of 
the  survey.  Care  should  be  taken  to  leave  no  spot  unex- 
plored. 

For  great  accuracy,  anchor  at  some  point,  and  determine 
its  place  as  above,  and  then  proceed  to  another  point,  paying 
out  a  line,  fastened  to  the  anchor,  and  sounding  at  regular 
distances.  Cast  anchor  at  the  second  point,  go  back  to  the 
first,  take  up  the  anchor,  go.  on  to  the  second,  and  then  pro- 
ceed as  before. 

The  soundings,  or  depth  of  the  water,  are  made  with  rods, 
chains,  or  lines,  according  to  the  depth  and  the  precision  re- 
quired. 


SOUNDINGS.  139 

(219.)  The  sounding-line  should  be  strong  and  pliable. 
The  lead  fastened  to  its  extremity  should  be  shaped  like  the 
frustum  of  a  cone.  The  size  of  the  line  and  the  weight  of  the 
lead  will  depend  upon  the  depth  of  the  water.  The  hand 
lead-line  is  limited  to  twenty  fathoms  and  is  marked  thus  :  at 
three  fathoms  a  piece  of  leather ;  at  five,  a  white  rag ;  at 
seven,  a  red  rag ;  at  ten,  a  piece  of  leather  with  a  round  hole 
in  it ;  at  thirteen,  a  blue  rag ;  at  fifteen,  a  white  rag ;  at  sev- 
enteen, a  red  rag  ;  at  twenty,  a  piece  of  cord  with  two  knots. 
These  divisions  are  called  marks.  The  other  divisions  called 
deeps  are :  at  half  a  fathom  a  piece  of  leather  with  three 
points ;  at  one  fathom  a  piece  of  leather  with  one  point ; 
at  two  fathoms  a  piece  of  leather  with  two  points  ;  at  four, 
six,  eight,  eleven,  fourteen,  sixteen,  and  eighteen  fathoms  a 
piece  of  cord  with  a  knot  in  it ;  at  nine  and  twelve, 
a  piece  of  cord  with  two  knots  in  it.  The  deep-sea  PIQ- 
lead-line  is  similarly  marked  up  to  twenty  fathoms. 
Each  additional  ten  fathoms  is  indicated  by  a  cord  with 
an  additional  knot,  and  half-way  between  these  a 
piece  of  leather  marks  the  five  fathoms. 

The  length  of  the  line  should  be  frequently  tested. 

The  character  of  the  bottom  is  determined  by  pla- 
cing tallow  into  a  hollow  in  the  base  of  the  lead,  which 
adheres  to  the  material  at  the  bottom.    A  barbed  pike 
is  sometimes  attached  to  the  base  of  the  lead  for  this  purpose.1 
Fig.  170. 

1  For  description  of  sounding  apparatus,  see  TT.  S.  C.  S.  Report,  1860 ;  for 
deep-sea  soundings,  Reports  of  1854,  1858,  and  1859. 


140      LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

CHAPTER  III. 

T  I  D  E- W  A  T  E  R  S. 

(220.)  The  soundings  taken  must  all  be  reduced  to  mean 
low  spring-tides. 

The  tides  are  semidiurnal  oscillations  of  the  ocean,  caused 
by  the  combined  attractions  of  the  moon  and  sun,  especially 
the  former.  The  tide  is  not  a  current,  but  a  broad,  flat  wave. 
The  lunar  one  follows  the  moon,  about  30°  east  of  it.  The 
solar  one  follows  the  sun.  In  the  open  sea,  it  is  high  water 
about  30°  east  of  the  moon.  To  the  east  of  this  the  tide  is 
ebbing  ;  west  of  this  it  is  rising ;  90°  distant  it  is  low  tide. 

The  tides,  being  caused  by  the  attraction  of  the  sun  and 
moon,  are  greatest  when  they  act  together ;  i.  e.,  at  new  and 
full  moon,  or  a  day  or  two  after.  These  are  spring-tides. 

They  are  least  when  the  sun  and  moon  act  perpendicularly 
to  each  other.  These  are  neap-tides.  The  spring-tides  are  the 
highest  and  lowest  tides.  The  moon  makes  a  tidal  wave  which 
recurs  in  12  hours  24  minutes,  while  the  sun's  recurs  in  12 
hours.  The  coincidence  of  these  waves  produces  spring-tides, 
and  their  partial  cancellation  neap-tides. 

If  the  tide- wave  met  with  no  obstruction,  the  highest  tides 
would  be  in  those  latitudes  over  which  the  sun  and  moon  pass 
vertically ;  but  the  height  of  the  tide  is  most  affected  by  local 
causes,  such  as  the  shoaling  of  the  water,  formation  of  the 
shore,  and  the  position  and  character  of  the  channels.  For 
exa'mple,  the  mean  height  of  the  tide  at  Cape  Florida  is  1.5 
feet,  while  in  the  Bay  of  Fundy  it  rises  over  40  feet. 

The  height  of  the  tide  is  also  affected  by  the  wind  and  the 
state  of  the  atmosphere.  A  wind  in  the  direction  in  which 
the  tide  is  moving,  and  a  low  barometer,  will  increase  the 
height,  and  vice  versa. 

(221.)  The  tide  caused  by  the  upper  transit  of  the  moon  is 
called  the  superior  tide,  and  that  caused  by  the  lower  transit 


TIDE-WATERS. 


141 


the  inferior  tide.  When  the  moon  is  north  of  the  equator, 
the  superior  tide  will  be  higher  than  the  inferior  tide.  This 
difference  in  the  height  of  the  tides  is  called  the  diurnal 
inequality.  On  the  Atlantic  coast,  the  successive  high  waters 
and  successive  low  waters  are  nearly  at  equal  heights  above 
and  below  the  mean,  with  intervals  of  12  hours  and  24 
minutes.  On  the  Pacific  coast,  the  successive  high  and  low 
tides  may  differ  several  feet  in  height,  and  several  hours  in 
intervals,  as  is  shown  in  Fig.  171 ;  a  and  c  are  successive  high 
tides,  and  5  and  d  low  tides. 


FIG.  171. 


The  lengthening  or  shortening  of  the  interval  between 
two  high  tides  is  called  the  priming  and  lagging  of  the  tide. 

(222.)  The  heights  of  mountains  and  other  points  on  the 
surface  of  the  earth  are  often  referred  to  the  level  of  the  sea 
as  a  datum  plane.  The  mean  level  of  the  sea  is  the  mean  be- 
tween the  mean  of  two  successive  high  tides  and  the  mean  of 
the  intermediate  low  tides.  This  is  constant,  while  high  and 
low  tides  vary. 

(223.)  The  tidal  current  in  channels  is  due  to  the  change 
of  level,  caused  by  the  tidal  wave.  The  rising  of  the  tide 
which  causes  the  flowing  in  of  the  current  is  called  flood-tide  ; 
and  the  falling  of  the  tide,  which  causes  the  flowing  out  of 
the  current,  is  called  ebb-tide.  The  stand  is  the  period  during 
which  the  height  of  the  tide  remains  stationary.  Slack  water 
is  the  interval  of  time  during  which  there  is  no  current. 
Where  a  narrow  inlet  connects  a  large  inland  basin  with  the 


14:2      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

sea,  care  should  be  taken  not  to  confound  the  time  of  high 
water  or  low  water  with  the  time  of  the  turn  of  the  tide.  The 
flood-tide  may  run  for  hours  after  the  time  of  high  water,  and 
so,  too,  the  stream  of  the  ebb-tide  after  low  water. 

(224.)  "  Establishment  of  the  Port."  Owing  to  the  ob- 
structions which  the  tidal  wave  meets  with  from  the  formation 
of  the  sea-bed  as  it  approaches  the  shore,  and  the  character 
and  direction  of  the  channels,  the  time  of  high  water  will 
differ  for  different  ports  in  the  same  vicinity.  In  order  that 
navigators,  entering  a  port,  may  be  able  to  find  the  time  of 
high  water,  a  standard  tide-time  is  established,  i.  e.,  the  num- 
ber of  hours  at  which  high  water  occurs  after  the  moon's 
transit  over  the  meridian.  This  is  called  the  "Establishment 
of  the  Port."  This  time  varies  with  the  age  of  the  moon. 
When  observed  on  the  days  of  full  or  change,  it  is  the  "  Vul- 
gar Establishment  of  the  Port."  The  "  Corrected  Establish- 
ment of  the  Port "  is  the  mean  of  the  intervals  between  the 
times  of  the  transit  of  the  moon  and  the  times  of  high  tide 
for  half  a  month.  This  is  used  for  finding  the  time  of  high 
water  on  any  given  day,  and  tables  are  constructed,  from 
observations  at  the  principal  posts,  for  finding  the  correction 
for  semi-monthly  inequality. 

(225.)  Tide-Gauges.  Tidal  observations  consist  in  recording 
the  heights  of  the  water  at  stated  times.  In  order  to  deter- 
mine this,  tide-gauges  are  necessary.  The  simplest  form  is  a 
stick  of  timber,  graduated  to  feet  and  inches,  or  tenths,  and 
either  set  up  in  the  water,  or  fastened  to  the  face  of  a  dock,  or 
pier,  so  that  the  rise  of  the  tide  may  be  noted  upon  it.  The 
zero-point  of  each  gauge  is  taken  at  or  below  the  lowest  tide, 
and  is  referred  to  a  permanent  "  bench-mark  "  on  the  shore. 
On  account  of  the  difficulty  of  sustaining  a  timber  of  consider- 
able height  against  the  force  of  the  wind  and  waves,  several 
successive  gauges  are  sometimes  used — the  bottom  mark  on 
each  gauge  higher  up  being  on  a  level  with  the  top  line  of  the 
next  lower.  Such  an  arrangement  is  required  on  gentle  slopes. 

On  the  sea-coast,  where  the  waves  make  the  reading  of  the 


TIDE-WATERS. 


143 


staff  difficult,  the  staff  may  be  attached  to  a  float,  enclosed  in 
an  upright  tube,  pierced  with  holes.  The  holes  in  the  tube 
should  be  of  such  a  size  as  to  allow  the  water  to  find  the  mean 
height  inside,  and  yet  reduce  the  oscillations  to  very  small 
limits.  Permanent  tide-gauges  should  be  self-registering.  For 
a  description  of  a  self-registering  tide-gauge,  see  U.  S.  C.  S. 
Keport,  1853. 

(226.)  Tide-Tables.  Observations  of  tides  may  be  recorded 
graphically,  as  on  page  143,*  or  in  the  tabular  form,  given  on 
page  144.*  In  the  table,  on  page  143,*  the  horizontal  line  is 
divided  into  months,  days,  and  half-days.  The  hours  and 
minutes  are  noted  at  the  feet  of  the  vertical  ordinates,  in 
black  for  high  water,  and  red  for  low  water.  The  lines  of  the 
ordinates  are  black  for  high  water,  and  red  for  low  water. 
The  heights  of  the  tides  are  noted  in  feet  and  tenths  at  the 
side  of  the  ordinates,  and  the  weather  at  their  summits.  An- 
other graphical  method  is  given  in  Fig.  171. 

The  tabular  form,  used  on  the  United  States  Coast  Survey, 
is  given  on  page  144.*  In  the  column  of  remarks,  the  posi- 
tion of  the  gauge  should  be  accurately  described,  and  the 
position  and  height  of  the  "  bench-mark,"  to  which  the  zero 
of  the  gauge  is  referred,  so  that  the  gauge  may  be  replaced  if 
disturbed. 

Table  of  records  of  tidal  observations  at  some  important 
points : 


Interval  between  time 

of    moon's    transit 

RISE  AND  PALL. 

MEAN  DURATION  OP 

(southing)  and  time 

STATION. 

of  high  water. 

Difference  be- 

Mean. 

tween  great- 
est and  least 

Mean. 

Spring. 

Nea^. 

Flood. 

Ebb. 

Stand. 

H.    M. 

H.    M. 

Feet. 

Feet. 

Feet. 

H.    M. 

H.    M. 

H.    M. 

Portland, 

11  25 

0  44 

8.8 

10.0 

7.6 

6  14 

6  12 

0  20 

Boston, 

11  22 

0  44 

10.1 

13.1 

7.4 

6  16 

6  18 

0  09 

New  York, 

8  13 

0  46 

4.3 

5.4 

3.4 

6  00 

6  25 

0  28 

Charlestown, 

7  13 

0  36 

5.3 

6.3 

4.6 

6  36 

6  09 

0  33 

Key  West, 

9  22 

1  12 

1.4 

2.3 

0.7 

6  59 

5  25 

0  12 

San  Francisco, 

12  03 

1  22 

3.9 

5.0 

2.9 

6  30 

5  52 

0  30 

144      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

(227.)  In  rivers,  a  number  of  tide-gauges  are  necessary,  at 
moderate  distances  apart,  especially  at  the  bends,  because  the 
tidal  lines  of  high  and  low  water  are  not  parallel  to  one 
another. 

The  soundings  are  to  be  reduced  by  the  nearest  gauge,  or 
by  the  mean  of  the  two  between  which  they  may  be  taken. 

(228.)  Beacons  and  Buoys.  Beacons  are  permanent  objects, 
such  as  piles  of  stones  with  signals  on  them,  usually  on  shoals 
and  dangerous  rocks. 

Buoys  are  floating  objects,  such  as  barrels,  or  hollow  iron 
spheres  or  cylinders,  anchored  by  a  chain,  and  variously 
painted,  to  indicate  either  dangers  or  channels. 
*  Those  placed  by  the  United  States  Coast  Survey  are  so 
colored  and  numbered  that  in  entering  a  bay,  harbor,  or  chan- 
nel, red  buoys  with  even  numbers  shall  be  passed  on  the  star- 
board or  right  hand,  black  buoys  with  odd  numbers  on  the 
port  hand  or  left  hand,  and  buoys  with  red  and  black  stripes, 
on  either  hand.  Buoys  in  channel-ways  are  colored  with 
alternate  white  and  black  vertical  stripes. 

On  dangerous  coasts,  self-ringing  bells  and  "  fog- whistles  " 
are  used. 


TIDE   WATERS, 


143* 


8TF 


6  £ 


TIP 


f 


Tcfc 


LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


TABULAR  FORM  FOR  TIDAL  OBSERVATIONS 


BAROM 
TER. 


MEAN 
O 
SERV 


•Jty 


•801 


•80JO,! 


V 


THE   CHART. 


CIIAPTEE  VI. 

THE   CHART. 

(229.)  HAVING  determined  the  lines  of  high  and  low  water, 
the  position  of  the  channels,  rocks,  shoals,  etc.,  and  the 
soundings,  a  chart  must  be  made,  on  which  all  these  are  laid 
down  in  their  proper  places.  For  scales  see  Art.  (171.) 

The  high-water  line  is  platted  like  FIG.  172. 

the  bounding  lines  of  a  farm.  The 
points  determined  in  the  low-water 
line,  and  the  positions  of  the  boat,  de- 
termined by  the  method  given  in  Art. 
(213),  are  fixed  on  the  chart  by  one 
of  the  methods  given  in  Arts.  (215), 
(216),  and  (217).  Contour  curves  are 
drawn  as  in  land  topography  (Part  -'*  ...*"'* 

IV.),  for  the  first  four  fathoms.  These 

may  be  indicated  by  dotted  lines,  as  in  Fig.  172,  or  they  may 
be  shaded  with  Indian-ink,  as  in  Fig.  173. 

FIG.  173. 


V 

»°    t 


Beyond  four  fathoms,  the  depths  are  noted  in  fathoms  and 
vulgar  fractions. 


146       LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 


(230.)  Various  conventional  signs  are  used  ;  some  of  the 
principal  ones  are  given  in  Figs.  174-194  : 


FIG.  174. 
Rocky  shore. 


FIG.  175. 
Rocks  always  bare. 


FIG.  176. 
Low,  swampy  shore. 


FIG.  179. 
Sandy  shore,  with  hillocks. 


I 1 


Anchorage  for  ships, 


186 


i 


Anchorage  for  coasters,       j    137 


Buoys. 


189 


Rocks  always  covered,       T 


Wrecks. 


Harbors. 


188 


Light-house,  <p>  192 

Signal-house,          K  193 

Channel-marks  ^^  <9  ^94 


PART  VIII. 
SPHERICAL  SURVEYING,  OR  GEODESY. 


CHAPTEK  I. 

THE      FIELD-WORK. 

(231.)  Nature.  It  comprises  the  methods  of  surveying  sur- 
faces of  such  extent  that  the  curvature  of  the  earth  cannot  be 
neglected.  The  method  of  triangulation  is  usually  employed. 

(232.)  TRIANGULAR  SURVEYING  is  founded  on  the  method 
of  determining  the  position  of  a  point  by  the  intersection  of 
two  known  lines.     Thus,  the  point  P 
is  determined  by  knowing  the  length 
of  the  line  A  B,  and  the  angles  P  B  A  /®\ 

and  P  A  B,  which  the  lines  P  A  and  /'       \ 

P  B  make  with  A  B.    By  an  extension  /* 

of  the  principle,  a  field,  a  farm,  or  a  %  /     \B 

country,  can  be  surveyed  by  measuring 

only  one  line,  and  calculating  all  the  other  desired  distances, 
which  are  made  sides  of  a  connected  series  of  imaginary  Tri- 
angles, whose  angles  are  carefully  measured.  The  district 
surveyed  is  covered  with  a  sort  of  net-work  of  such  triangles, 
whence  the  name  given  to  this  kind  of  surveying.  It  is  more 
commonly  called  "Trigonometrical  Surveying,"  and  some- 
times "  Geodesic  Surveying,"  but  improperly,  since  it  does 
not  necessarily  take  into  account  the  curvature  of  the  earth, 
though  always  adopted  in  the  great  surveys  in  which  that  is 
considered. 


LEVELLING,  TOPOGRAPHY,  AND   HIGHER   SURVEYING. 

(233.)  Outline  of  Operations.  A  lase-line,  as  long  as  possi- 
ble (fiVe  or  ten  miles  in  surveys  of  countries),  is  measured  with 
extreme  accuracy. 

From  its  extremities,  angles  are  taken  to  the  most  distant 
objects  visible,  such  as  steeples,  signals  on  mountain-tops,  etc. 

The  distances  to  these  and  between  these  are  then  calcu- 
lated by  the  rules  of  trigonometry. 

The  instrument  is  then  placed  at  each  of  these  new  sta- 
tions, and  angles  are  taken  from  them  to  still  more  distant 
stations,  the  calculated  lines  being  used  as  new  base-lines. 

This  process  is  repeated  and  extended  till  the  whole  dis- 
trict is  embraced  by  these  "  primary  triangles "  of  as  large 
sides  as  possible. 

One  side  of  the  last  triangle  is  so  located  that  its  length 
can  be  obtained  by  measurement  as  well  as  by  calculation, 
and  the  agreement  of  the  two  proves  the  accuracy  of  the 
whole  work. 

Within  these  primary  triangles,  secondary  or  smaller  tri- 
angles are  formed,  to  fix  the  position  of  the  minor  local  details, 
and  to  serve  as  starting-points  for  common  surveys  with  chain 
and  compass,  etc.  Tertiary  triangles  may  also  be  required. 

The  larger  triangles  are  first  formed,  and  the  smaller  ones 
based  on  them,  in  accordance  with  the  important  principle  in 
all  surveying  operations,  always  to  work  from  the  whole  to 
the  parts,  and'  from  greater  to  less. 

(234.)  Measuring  a  Base.  Extreme  accuracy  in  this  is  neces- 
sary, because  any  error  in  it  will  be  multiplied  in  the  sub- 
sequent work.  The  ground  on  which  it  is  located  must  be 
smooth  and  nearly  level,  and  its  extremities  must  be  in  sight 
of  the  chief  points  in  the  neighborhood.  Its  point  of  begin- 
ning must  be  marked  by  a  stone  set  in  the  ground  with  a  bolt 
let  into  it.  Over  this  a  theodolite  or  transit  is  to  be  set,  and 
the  line  "  ranged  out."  The  measurement  may  be  made  with 
chains  (which  should  be  formed  like  that  of  a  watch),  etc.,  but 
best  with  rods.  We  will  notice,  in  turn,  their  materials,  sup- 
ports, alignement,  levelling,  and  contact. 


THE  FIELD-WORK.  149 

As  to  materials,  iron,  brass,  and  other  metals,  have  been 
used,  but  are  greatly  lengthened  and  shortened  by  changes  of 
temperature.  Wood  is  affected  by  moisture.  Glass  rods  and 
tubes  are  preferable  on  both  these  accounts.  But  wood  is  the 
most  convenient.  "Wooden  rods  should  be  straight-grained 
white  pine,  etc.,  well  seasoned,  baked,  soaked  in  boiling  oil, 
painted  and  varnished.  They  may  be  trussed,  or  framed  like 
a  mason's  plumb-line  level,  to  prevent  their  bending.  Ten  or 
fifteen  feet  is  a  convenient  length.  Three  are  required,  which 
may  be  of  different  colors,  to  prevent  mistakes  in  recording. 
They  must  be  very  carefully  compared  with  a  standard  measure. 

Supports  must  be  provided  for  the  rods,  in  accurate  work. 
Posts,  set  in  line  at  distances  equal  to  the  length  of  the  rods, 
may  be  driven  or  sawed  to  a  uniform  line,  and  the  rods  laid 
on  them,  either  directly  or  on  beams  a  little  shorter.  Tripods, 
or  trestles,  with  screws  in  their  tops  to  raise  or  lower  the  ends 
of  the  rods 'resting  on  them,  or  blocks  with  three  long  screws 
passing  through  them  and  serving  as  legs,  may  also  be  used. 
Staves,  or  legs,  for  the  rods  have  been  used,  these  legs  bearing 
pieces  which  can  slide  up  and  down  them,  and  on  which  the 
rods  themselves  rest. 

The  alignement  of  the  rods  can  be  effected,  if  they  are  laid 
on  the  ground,  by  strings,  two  or  three  hundred  feet  long, 
stretched  between  the  stakes  set  in  the  line,  a  notched  peg  be- 
ing driven  when  the  measurement  has  reached  the  end  of  one 
string,  which  is  then  taken  on  to  the  next  pair  of  stakes  ;  or, 
if  the  rods  rest  on  supports,  by  projecting  points  on  the  rods 
being  aligned  by  the  instrument. 

The  levelling  of  the  rods  can  be  performed  with  a  common 
mason's  level ;  or  their  angle  measured,  if  not  horizontal,  by 
a  "  slope-level." 

The  contacts  of  the  rods  may  be  effected  by  bringing  them 
end  to  end.  The  third  rod  must  be  applied  to  the  second  be- 
fore the  first  has  been  removed,  to  detect  any  movement.  The 
ends  must  be  protected  by  metal,  and  should  be  rounded  (with 
radius  equal  to  length  of  rod),  so  as  to  touch  in  only  one  point. 
Round-headed  nails  will  answer  tolerablv.  Better  are  small 


150       LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

steel  cylinders,  horizontal  on  one  end  and  vertical  on  the 
other.  Sliding  ends,  with  verniers,  have  been  used.  If  one 
rod  be  higher  than  the  next  one,  one  must  be  brought  to  touch 
a  plumb-line  which  touches  the  other,  and  its  thickness  be 
added.  To  prevent  a  shock  from  contact,  the  rods  may  be 
brought  not  quite  in  contact,  and  a  wedge  be  let  down  be- 
tween them  till  it  touches  both  at  known  points  on  its  gradu- 
ated edges.  The  rods  may  be  laid  side  by  side,  and  lines 
drawn  across  the  end  of  each  be  made  to  coincide  or  form  one 
line.  This  is  more  accurate.  Still  better  is  a  "  visual  con- 
tact," a  double  microscope  with  cross-hairs  being  used,  so 
placed  that  one  tube  bisects  a  dot  at  the  end  of  one  rod,  and 
the  other  tube  bisects  a  dot  at  the  end  of  the  next  rod.  The 
rods  thus  never  touch.  The  distance  between  the  two  sets  of 
cross-hairs  is  of  course  to  be  added. 

A  base  could  be  measured  over  very  uneven  ground,  or 
even  water,  by  suspending  a  series  of  rods  from  a  stretched 
rope  by  rings  in  which  they  can  move,  and  levelling  them  and 
bringing  them  into  contact  as  above. 

The  most  perfect  base-measuring  apparatus  is  that  used  on 
the  United  States  Coast  Survey.1  It  consists  of  a  bar  of  brass 
and  a  bar  of  iron,  a  little  less  than  six  metres  long,  supported 
parallel  to  each  other,  firmly  attached  to  a  block  at  one  end, 
and  left  free  to  move  at  the  other,  so  that  the  entire  contrac- 
tion and  expansion  are  at  that  end.  At  right  angles  to  these 
bars  is  a  short  lever,  called  the  "  lever  of  compensation."  It 
is  attached  to  the  lower  (brass)  bar  at  the  free  end  by  a  hinge, 
and  an  agate  knife-edge  on  the  lever  rests  against  a  steel  plate 
at  the  end  of  the  iron  bar. 

When  the  temperature  is  raised,  both  bars  expand,  but  the 
brass  one  more  than  the  iron  one,  so  that  the  upper  end  of  the 
lever  of  compensation  is  thrown  back.  A  knife-edge,  turned 
outward,  is  placed  on  the  lever,  at  such  a  distance  from  the 
other  knife-edge  and  the  hinge,  that  it  shall  remain  unmoved 
by  equal  changes  of  temperature  in  the  two  bars. . 

Brass  and  iron,  exposed  to  the  same  temperature,  will  not 

1  For  a  full  description,  see  Coast  Survey  Report  of  1854. 


THE  FIELD-WORK.  151 

heat  equally  in  equal  times.  To  overcome  this  difficulty,  the  bars 
are  given  equal  absorbing  surfaces,  but  their  cross-sections  are 
adapted  to  their  different  specific  heats  and  conducting  powers. 

The  knife-edge  on  the  upper  end  of  the  lever  of  compen- 
sation presses  against  a  short  sliding  rod,  supported  on  the  upper 
(iron)  bar,  and  held  firmly  against  the  lever  by  a  spiral  spring. 
The  sliding  rod  is  terminated  on  the  outer  end  by  an  agate  plane. 

The  end  of  the  apparatus  we  have  been  considering  is 
called  the  compensating  end.  "We  will  now  consider  the  sec- 
tor end,  where  are  arranged  the  parts  for  adjusting  the  con- 
tacts between  the  successive  rods  in  measuring ;  and  for  de- 
termining the  inclination  of  the  rod  on  sloping  ground. 

This  end  also  terminates  in  a  sliding  rod,  bearing  on  its 
extremity  an  agate  knife-edge,  placed  horizontally,  and  resting 
by  its  inner  end  against  an  upright  "  lever  of  contact."  This 
lever  is  fastened  by  a  hinge  at  the  lower  end,  and  its  upper 
end  rests  against  a  tongue,  attached  to  the  "  level  of  contact," 
which  is  mounted  on  trunnions.  "When  the  sliding  rod  is 
moved  in,  the  lever  of  contact  presses  against  the  tongue  of 
the  level  of  contact  and  turns  the  level.  The  inner  end  of  the 
level-tube  is  weighted  so  as  to  insure  a  constant  pressure  when 
the  contact  is  made  between  two  rods,  and  the  bubble  is 
brought  to  the  centre.  The  sector  is  an  arrangement  for  de- 
termining the  angle  at  which  the  rod  is  inclined. 

The  whole  apparatus  is  enclosed  in  a  double  tin  tubular 
case,  only  the  ends  of  the  sliding  rods,  bearing  the  agates, 
being  exposed.  The  observations  are  taken  through  glass 
doors  in  the  side  of  the  tube.  The  extreme  length  is  six  me- 
tres. Two  of  these  tubes  are  used  in  measuring  a  base,  and 
each  is  supported  by  two  trestles.  The  tubes  are  aligned  by 
the  aid  of  a  transit. 

On  one  base,  seven  miles  long,  measured  with  this  appara- 
tus, the  greatest  supposable  error  was  computed,  from  remeas- 
urements,  to  be  less  than  six-tenths  of  an  inch.  On  another 
base,  six  and  three-quarter  miles  long,  the  probable  error  was 
less  than  one-tenth  of  an  inch,  and  the  greatest  supposable 
error  was  less  than  three-tenths  of  an  inch. 


152       LEVELLING,  TOPOGRAPHY,  AND   HIGHER  SURVEYING. 

(235.)  Corrections  of  Base,  If  the  rods  were  not  level,  their 
length  must  be  reduced  to  its  horizontal  projection.  This 
would  be  the  square  root  of  the  difference  of  the  squares  of 
the  length  of  the  rod  (or  of  the  base),  and  of  the  height  of  one 
end  above  the  other  ;  or  the  product  of  the  same  length  by 
the  cosine  of  the  angle  which  it  makes  with  the  horizon.1 

If  the  rods  were  metallic,  they  would  need  to  be  corrected 
for.  temperature.  Thus,  if  an  iron  bar  expands  Tinrjlnnr  of  its 
length  for  1°  Fahr.,  and  had  been  tested  at  32°,  and  a  base 
had  been,  measured  at  72°  with  such  a  bar  10  feet  long,  and 
found  to  contain  3,000  of  them,  its  apparent  length  would  be 
30,000  feet,  but  its'real  length  would  be  SA  feet  more. 

EXPANSION   FOR    1°    FAHRENHEIT. 

Brass  bar  =  0.00001050903  ; 

Iron  bar  =  0.000006963535  ; 
Platinum  =  0.0000051344; 
Glass  =  0.0000043119  ; 
White  Pine  =  0.0000022685. 

(236.)  Reducing  the  Base  to  the  Level  of  the  Sea.      Let 

A  B  =  a  be  the  measured  base,  and 

A'  B'  =  a?,  the  base  reduced  to  the 

level  of  the  sea,  h  the  height  of  the 

measured  base  above  the  level  of  the 

sea,  and  r  the  radius  of  the  earth  to 

the  level  of  the  sea.    Then  we  have  : 

T  +  h  :  r  ::  a  :  x 

r 


ah         r       ah 


1  More  precisely,  A  being  this  angle,  and  not  more  than  2°  or  3°,  the  difference 
between  the  inclined  and  horizontal  lengths  equals  the  inclined  or  real  length  mul- 
tiplied by  the  square  of  the  minutes  in  A,  and  that  by  the  decimal  0.00000004231. 


TV 


THE  FIELD-WORK.  153 

Developing  by  the  binomial  formula,  we  get : 

li          h*         ha 
a  —  x  =  a 0  -5  4*  #  "i  — ,  etc. 

/V*  ft*  /Y>  * 

0 

As  h  is  very  small  in  comparison  with  r,  the  first  term  of  the 
correction  is  generally  sufficient. 

(237.)  A  Broken  Base.     When  the  angle  C  is  very  obtuse, 
the  lines  A  C  and  C  B  being  measured,  and  forming  nearly  a 

Fia.  197. 


straight  line,  the  length  of  the  line  A  B  is  found  thus :  Naming 
the  lines,  as  is  usual  in  trigonometry,  by  small  letters  cor- 
responding to  the  capital  letters  at  the  angles  to  which  they 
are  opposite,  and  letting  K  =  the  number  of  minutes  in  the 
supplement  of  the  angle  C,  we  shall  have : 

AB  =  c  —  a  +  l  —  0.000000042308  x  ^— -*  . 

a  +  b 

Log.  0.000000042308  =  2.6264222  -  10. 

(238.)  Base  of  Verification.  As  mentioned  in  Art.  (233),  a 
side  of  the  last  triangle  is  so  located  that  it  can  be  measured, 
as  was  the  first  base.  If  the  measured  and  calculated  lengths 
agree,  this  proves  the  accuracy  of  all  the  previous  work  of 
measurement  and  calculation,  since  the  whole  is  a  chain  of 
which  this  is  the  last  link,  and  any  error  in  any  previous  part 
would  affect  the  very  last  line,  except  by  some  improbable 
compensation.  How  near  the  agreement  should  be,  will  de- 
pend on  the  nicety  desired  and  attained  in  the  previous  opera- 
tions. Two  bases,  60  miles  distant,  differed  on  one  great 
English  survey  28  inches ;  on  another,  1  inch ;  and  on  a  French 


154      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

triangulation  extending  over  500  miles,  the  difference  was  less 
than  2  feet.  Results  of  equal  or  greater  accuracy  are  obtained 
on  the  United  States  Coast  Survey.  "  The  Fire  Island  base, 
on  the  south  side  of  Long  Island,  and  the  Kent  Island  base  in 
Chesapeake  Bay,  are  connected  by  a  primary  triangulation. 
This  Kent  Island  base  is  5  miles  and  4=  tenths  long,  and  the 
original  Fire  Island  base  is  8  miles  and  7  tenths.  The  short- 
est distance  between  them  is  208  miles,  but  the  distance 
through  the  triangulation  is  320.  The  number  of  intervening 
triangles  is  32,  yet  the  computed  and  measured  lengths  of  the 
Kent  Island  base  exhibit  a  discrepancy  no  greater  than  4 
inches." 

(239.)  Choice  of  Stations.  The  stations,  or  "trigonometri- 
cal points,"  which  are  to  form  the  vertices  of  the  triangles, 
and  to  be  observed  to  and  from,  must  be  so  selected  that 
the  resulting  triangles  may  be  "  well-conditioned,"  i.  e.,  may 
have  such  sides  and  angles  that  a  small  error  in  any  of  the 
measured  quantities  will  cause  the  least  possible  errors  in  the 
quantities  calculated  from  them.  The  higher  calculus  shows 
that  the  triangles  should  be  as  nearly  equilateral  as  possible. 
This  is  seldom  attainable,  but  no  angle  should  be  admitted 
less  than  30°,  or  more  than  120°.  When  two  angles  only 
are  observed,  as  is  often  the  case  in  the  secondary  triangu- 
lation, the  unobserved  angle  ought  to  be  nearly  a  right 
angle. 

To  extend  the  triangulation^  by  continually  increasing  the 
sides  of  the  triangles,  without  introducing  "  ill-conditioned  " 
triangles,  may  be  effected  as  in  Fig.  198.  A  B  is  the  measured 
base,  C  and  D  are  the  nearest  stations.  In  the  triangles  ABC 
and  A  B  D,  all  the  angles  being  observed,  and  the  side  A  B 
known,  the  other  sides  can  be  readily  calculated.  Then,  in 
each  of  the  triangles  D  A  C  and  D  B  C,  two  sides  and  the 
contained  angles  are  given  to  find  D  C,  one  calculation  check- 
ing the  other.  DC  then  becomes  a  base  to  calculate  EF, 
which  is  then  used  to  find  G  H,  and  so  on. 

The  fewer  primary  stations  used  the  better,  both  to  pre- 


THE  FIELD-WORK. 


155 


vent  confusion  and  because  the  smaller  number  of  triangles 
makes  the  correctness  of  the  results  more  "  probable." 


The  United  States  Coast  Survey,  under  the  superintend- 
ence of  Prof.  A.  D.  Bache,  displays  some  fine  illustrations  of 
these  principles,  and  of  the  modifications  they  may  undergo 
to  suit  various  localities.  The  figure  on  the  next  page  repre- 
sents part  of  the  scheme  of  the  primary  triangulation  resting 
on  the  Massachusetts  base,  and  including  some  remarkably 
well-conditioned  triangles,  as  well  as  the  system  of  quadri- 
laterals, which  is  a  valuable  feature  of  the  scheme  when  the 
sides  of  the  triangles  are  extended  to  considerable  lengths, 
and  quadrilaterals,  with  both  diagonals  determined,  take  the 
place  of  simple  triangles. 

The  engraving  is  on  a  scale  of  1 : 1200,000. 


156      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

Fig.  199. 


o 
o 


THE  FIELD-WORK, 


157 


FIG.  200. 


(240.)  Signals.     They  must  be  high,  conspicuous,  and  so 
made  that  the  instrument  can  be  placed  precisely  under  them. 

Three  or  four  timbers  framed  into  a 
pyramid,  as  in  Fig.  200,  with  a  long  mast 
projecting  above,  fulfil  the  first  and  last 
conditions.  The  mast  may  be  made  ver- 
tical by  directing  two  theodolites  to  it,  and 
adjusting  it  so  that  their  telescopes  follow 
it  up  and  down,  their  lines  of  sight  being 
at  right  angles  to  each  other.  Guy-ropes 
may  be  used  to  keep  it  vertical. 

A  very  excellent  signal,  used  on  the 
Massachusetts  State  Survey,  by  Mr.  Borden,  is  represented  in 
the  three  following  figures.     It  consists  merely  oi  three  stout 


FIG.  201. 


FIG.  202. 


FIG.  203. 


FIG.  204. 


sticks,  which  form  a  tripod,  framed  with  the  signal-staff,  by  a 
bolt  passing  through  their  ends  and  its  middle.  Fig.  201 
represents  the  signal  as  framed  on  the  ground ;  Fig.  202  shows 
it  erected  and  ready  for  observation,  its  base  being  steadied 
with  stones ;  and  Fig.  203  shows  it  with  the  staff 
turned  aside,  to  make  room  for  the  theodolite  and 
its  protecting  tent.  The  heights  of  these  signals 
varied  between  15  and  80  feet. 

Another  good  signal  consists  of  a  stout  post  let 
into  the  ground,  with  a  mast  fastened  to  it  by  a 
bolt  below  and  a  collar  above.  By  opening  the 
collar,  the  mast  can  be  turned  down  and  the  the- 
odolite set  exactly  under  the  former  summit  of  the 
signal,  i.  e.,  in  its  vertical  axis. 


158      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

A  tripod  of  gas-pipe  has  been  used  to  support  the  signal  in 
positions  exposed  to  the  sea,  as  on  shoals.  It  is  taken  to  the 
desired  spot  in  pieces,  and  there  screwed  together  and  set  up. 

Signals  should  have  a  height  equal  to  at  least  ^oVo  °f  their 
distance,  so  as  to  subtend  an  angle  of  half  a  minute,  which 
experience  has  shown  to  be  the  least  allowable. 

To  make  the  tops  of  the  signal-masts  conspicuous,  flags 
may  be  attached  to  them  ;  white  and  red,  if  to  be  seen  against 
the  ground,  and  red  and  green  if  to  be  seen  against  the  sky.1 
The  motion  of  flags  renders  them  visible,  when  much  larger 
motionless  objects  are  not.  But  they  are  useless  in  calm 
weather.  A  disk  of  sheet-iron,  with  a  hole  in  it,  is  very  con- 
spicuous. It  should  be  arranged  so  as  to  be  turned  to  face 
each  station.  A  barrel,  formed  of  muslin  sewed  together,  four 
or  five  feet  long,  with  two  hoops  in  it  two  feet  apart,  and  its 
loose  ends  sewed  to  the  signal-staff,  which  passes  through  it,  is 
a  cheap  and  good  arrangement.  A  tuft  of  pine-boughs  fast- 
ened to  the  top  of  the  staff,  will  be  well  seen  against  the  sky. 

In  sunshine  a  number  of  pieces  of  tin,  nailed  to  the  staff  at 
different  angles,  will  be  very  conspicuous.  A  truncated  cone 
of  burnished  tin  will  reflect  the  sun's  rays  to  the  eye  in  almost 
every  situation. 

The  most  perfect  arrangement  is  the  "heliotrope,"  in- 
vented by  Gauss.  This  consists  of  a  mirror  a  few  inches 
square,  so  mounted  on  a  telescope,  near  the  eye-end,  that  the 
reflection  of  the  sun  may  be  thrown  in  any  desired  direction. 
They  have  been  observed  on  at  a  distance  of  80  or  90  miles, 
when  the  outlines  of  the  mountains  on  which  they  were  placed 
were  invisible.  A  man,  called  a  "  heliotroper,"  is  stationed 

1  To  determine  at  a  station  A, 
whether  its  signal  can  be  seen  from 
B,  projected  against  the  sky  or  not, 
measure  the  vertical  angles  B  A  Z 
and  Z  A  C.  If  their  sum  equals  or 
exceeds  180°,  A  will  be  thus  seen 
from  B.  If  not,  the  signal  at  A 
must  be  raised  till  this  sum  equals 
180°. 


THE  FIELD-WORK. 


159 


at  the  instrument.  He  directs  the  telescope  toward  the  sta- 
tion at  which  the  transit  is  placed  for  observation,  and  keeps 
the  mirror  turned  so  as  to  reflect  the  sun  in  a  direction  parallel 
to  the  axis  of  the  instrument.  This  he  accomplishes  by  caus- 
ing the  reflection  to  pass  through  two  perforated  disks,  mounted 
on  the  telescope,  one  near  the  object-end,  and  the  other  near 
the  mirror. 

For  night-signals,  an  Argand  lamp  is  used ;  or,  best  of 
all,  Drummond's  light,  produced  by  a  stream  of  oxygen  gas 
directed  through  a  flame  of  alcohol  upon  a  ball  of  lime.  Its 
distinctness  is  exceedingly  increased  by  a  parabolic  reflector 
behind  it,  or  a  lens  in  front  of  it.  Such  a  light  was  brilliantly 
visible  at  66  miles'  distance. 


(241.)  Observations  of  the  Angles.  These  should  be  repeated 
as  often  as  possible.  In  extended  surveys,  three  sets,  of  ten 
each,  are  recommended.  They  should  be  taken  on  different 
parts  of  the  circle.  In  ordinary  surveys,  it  is  well  to  employ 
the  method  of  "traversing,"  Art.  (94).  In  long  sights,  the 
state  of  the  atmosphere  has  a  ,  very  remarkable  effect  on  both 
the  visibility  of  the  signals  and  on  the  correctness  of  the  ob- 
servations. 

"When  many  angles  are  taken  from  one  station,  it  is  im- 
portant to  record  them  by  some  uniform  system.  The  form 
given  below  is  convenient.  It  will  be  noticed  that  only  the 
minutes  and  seconds  of  the  second  vernier  are  employed,  the 
degrees  being  all  taken  from  the  first : 


Observations  at 


Station 
observed  to. 

READINGS. 

Mean 
Reading. 

Right  or  Left 
of  PreQed- 
ing  Object. 

Remarks. 

Vernier  A. 

Vernier  B. 

A 

70°  19'    0" 

18'  40" 

70°  18'  50* 

B 

103°  32'  20" 

32'  40" 

103°  32'  30' 

R. 

C 

115°  14'  20" 

14'  50" 

115°  14'  35" 

R. 

11 


160      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

When  the  angles  are  "  repeated,"  the  multiple  arcs  will 
be  registered  under  each  other,  and  the  mean  of  the  seconds 
shown  by  all  the  verniers  at  the  first  and  last  readings  be 
adopted. 

THE  GKEAT  THEODOLITE,  used  on  the  Coast  Survey  for  the 
observation  of  the  angles  in  the  primary  triangles,  has  a 
horizontal  circle  thirty  inches  in  diameter,  graduated  to  five 
minutes,  and  reading  to  single  seconds  by  three  micrometer 
microscopes,  placed  120°  apart.  The  telescope  has  a  focal 
length  of  four  feet. 

When  the  country  over  which  the  triangulation  extends  is 
fiat,  it  has  been  found  necessary  to  elevate  the  transit  some 
distance  from  the  surface  of  the  ground,  the  stratum  of  air 
near  the  surface  being  so  disturbed  by  exhalations  and  ine- 
qualities of  temperature  and  density  as  to  render  accurate 
observation  impossible.  The  plan  adopted  on  the  Coast  Sur- 
vey is  as  follows :  On  the  top  of  a  signal-tripod,  forty-three 
feet  high,  is  placed  a  cap-block,  into  which  is  mortised  a  square 
hole  to  receive  the  signal-pole.  Around  the  tripod,  but  not 
touching  it,  is  erected  a  rectangular  scaffold,  forty  feet  high. 
On  the  top  of  it  is  a  platform,  from  which  the  observations 
are  taken,  the  signal-pole  being  removed  from  the  cap-block, 
and  the  transit  placed  so  that  its  centre  shall  be  precisely  over 
the  station-point. 

(242.)  Reduction  to  the  Centre.    It   is   often  impossible  to 
set  the  instrument  precisely  at  or  under  the  signal  which  has 
FIG.  206.  been  observed.    In  such  cases 

proceed  thus :  Let  C  be  the 
centre  of  the  signal,  and  R  C  L 
the  desired  angle,  R  being  the 

E^^^"^--..  right-hand  object  and  L  the 

left-hand  one.  Set  the  instru- 
ment at  D,  as  near  as  possible 
to  C,  and  measure  the  angle  R  D  L.  It  may  be  less  than 
R  C  L,  or  greater  than  it,  or  equal  to  it,  according  as  D  lies 
without  the  circle  passing  through  C,  L,  and  R,  or  within  it, 


THE   FIELD-WORK. 

or  in  its  circumference.  The  instrument  should  be  set  as 
nearly  as  possible  in  this  last'  position.  To  find  the  proper 
correction  for  the  observed  angle,  observe  also  the  angle  L  D  C 
(called  the  angle  of  direction),  counting  it  from  0°  to  360°, 
going  from  the  left-hand  object  toward  the  left,  and  measure 
the  distance  D  C.  Calculate  the  distances  C  R  and  C  L  with 
the  angle  R D  L,  instead  of  R  C  L,  since  they  are  sufficiently 
nearly  equal.  Then, 

CD.sin.(RDL+LDC)     C  D  .  sin.  L  D  C 

' — T^TTT -t  it —   T^Tr .        ~T7> 

OR.  sin.  1"  CL  .  sin.  V 

The  last  two  terms  will  be  the  number  of  seconds  to  be 
added  or  subtracted.  The  trigonometrical  signs  of  the  sines 
must  be  attended  to.  The  log.  sin.  V  —  4.6855749.  Instead 
of  dividing  by  sin.  1",  the  correction  without  it,  which  will  be 
a  very  small  fraction,  may  be  reduced  to  seconds  by  multiply- 
ing it  by  206265. 

Example.— Let  R  D  L  =  32°  20'  18".06 ;  L  D  C  =  101°  15' 
32".4;  CD  =  0.9;  CR  =  35845.12;  CL  =  29783.1. 

The  first  term  of  the  correction  will  be  +  3/r.750,  and  the 
second  term  —  6".113.  Therefore,  the  observed  angle  R  D  L 
must  be  diminished  by  2.".363,  to  reduce  it  to  the  desired  an- 
gle RCL. 

Much  calculation  may  be  saved  by  taking  the  station  D  so 
that  all  the  signals  to  be  observed  can  be  seen  from  it.  Then 
only  a  single  distance  and  angle  of  direction  need  be  measured. 

It  may  also  happen  that  the  centre,  C,  of  the 
signal  cannot  be  seen  from  D.     Thus,  if  the  sig-     ^0^^. 
nal  be  a  solid  circular  tower,  set  the  theodolite 
at  D,  and  turn  its  telescope  so  that  its  line  of 
sight  becomes  tangent  to  the  tower  at  T,  T' ; 
measure  on  these  tangents  equal  distances,  D  E, 
D  F,  and  direct  the  telescope  to  the  middle,  G, 
of  the  line  E  F.     It  will  then  point  to  the  centre, 
C ;  and  the  distance  D  C  will   equal  the  distance  from  D 
to  the  tower  plus  the  radius  obtained  by  measuring  the  cir- 
cumference. 


162      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

If  the  signal  be  rectangular,  measure  D  E,  D  F.     Take 

FIG  208.       an7  Pomt  G"  on  D  E>  and  on  D  F  set  off  D  H 

DF 
=  D  G  -Fr-p  •    Then  is  G  H  parallel  to  E  F  (since 


D  G  :  D  H  :  :  D  E  :  D  F),  and  the  telescope  di- 
rected to  its  middle,  K,  will  point  to  the  middle 
of  the  diagonal  E  F.  We  shall  also  have  D  0 


Any  such  case  may  be  solved  by  similar  methods. 

The  "phase  "  of  objects  is  the  effect  produced  by  the  sun 
shining  on  only  one  side  of  them,  so  that  the  telescope  will  be 
directed  from  a  distant  station  to  the  middle  of  that  bright 
side  instead  of  to  the  true  centre.  It  is  a  source  of  error  to 
be  guarded  against.  Its  effect  may,  however,  be  calculated. 

When  the  signal  is  a  tin  cone  : 

Let  T  =  radius  pf  the  signal, 

Z  =  angle  at  the  point  of  observation  between  the 

sun  and  the  signal, 
D  =  the  distance. 


Then,  the  correction  =  ± 


sin. 


(243.)  The  Angles.  The  triangles  observed  are  supposed  to 
have  sides  of  such  length  that  the  sum  of  the  three  angles  ex- 
ceeds 180°  by  a  certain  sensible  quantity  called  the  "  spherical 
excess"  This  is  usually  only  a  few  seconds.  For  a  triangle 
containing  about  76  square  miles,  which,  if  equilateral,  would 
have  sides  13  miles  long,  the  spherical  excess  is  only  one  sec- 
ond. For  a  triangle  with  sides  of  102  miles  it  is  one  minute. 

It  must  be  determined  before  we  can  know  how  much  the 
eiTor  is,  and  therefore  what  the  correct  sum  and  correction 
should  be. 

(244.)  The  true  spherical  excess  is  found  by  this  principle  : 
"  The  surface  of  a  spherical  triangle  is  measured  by  the  excess 


THE  FIELD-WORK.  1(33  • 

of  its  angles  above  two  right  angles  multiplied  by  the  trirec- 
tangular triangle."  1 

Hence  the  surfaces  of  spherical  triangles  are  to  each  other 
as  their  respective  spherical  excesses. 

Let  s  =  surface  of  given  triangle, 

t  =  surface  of  trirectangular  triangle, 
e  =  spherical  excess  of  given  triangle, 
ef=  spherical  excess  of  trirectangular  triangle. 
Then,  we  have : 

s  :  t  :  :  e  :  e' . 

t  =  %  surface  of  sphere  =  -J-  x  4  nr*  =  I  nr*. 
e'=  (8  x  90°)  -  180°  =  90°. 

Then,  s  :  ±1*1*  :  :  e  :  90°. 

_  648000" . 

YY  hence,  e  =  s  x  — -— ^ —  in  seconds. 

s  and  r  are  in  the  same  unit  of  measure. 

The  fraction  is  a  constant  quantity  whose  logarithm  is 
10.6746069,  the  mean  radius  of  the   earth   being  taken  a&  - 
20888629  feet ;  the  greater  radius  being  20923596,  and  the 
smaller  radius  20853662.' 

The  surface  s,  being  very  small  compared  with  r2,  may  be 
obtained  with  sufficient  accuracy  for  this  object  by  treating 
the  triangle  as  if  it  were  plane. 

Then,  when  two  sides  and  the  contained  angle  are  given, 
we  have : 

s  =  £  a  I  .  sin.  C. 

When  two  angles  and  the  included  side  are  given,  we  have : 

sin.  B  .  sin.  C 


8  = 


sin.  (B  +  C)  * 


Approximately,  the  spherical  excess  (in  seconds)  equals 
the  area  (in  square  miles)  divided  by  75.5. 

1  Davies's  legendre,  Book  IX.,  prop.  18.         *  According  to  Sir  John  HerseheL 

l~         I  ^>  {V      *^^» 

fl  N 

0     —         V    V*    "^  i  /  y  /£--£?  r  **»    *~\    /  / 

(j^  -^      o/^o  •*•  Xr*1  "2     *   <C 

I  I  1.       .  »      I  ^    '"  NX     4 


164       LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

Having  found  the  spherical  excess,  if  the  sum  of  the  angles 
of  the  triangle  does  not  equal  180°  plus  this  excess,  the  differ- 
ence is  distributed  among  them  as  in  Art.  (245). 

(245.)  Correction  of  the  Angles.  When  all  the  angles  of  any 
triangle  can  be  observed,  their  sum  should  equal  180°  plus  the 
"  spherical  excess."  If  not,  they  must  be  corrected.  If  all 
the  observations  are  considered  equally  accurate,  one-third  of 
the  difference  of  their  sum  from  180°  plus  the  spherical  excess, 
is  to  be  added  to  or  subtracted  from  each  of  them.  But  if  the 
angles  are  the  means  of  unequal  numbers  of  observations,  their 
errors  may  be  considered  to  be  inversely  as  those  numbers, 
and  they  may  be  corrected  by  this  proportion :  As  the  sum  of 
the  reciprocals  of  each  of  the  three  numbers  of  observations  is 
to  the  whole  error,  so  is  the  reciprocal  of  the  number  of  obser- 
vations of  one  of  the  angles  to  its  correction. 

It  is  still  more  accurate,  but  laborious,  to  apportion  the 
total  error,  or  difference  from  180°  plus  the  spherical  excess, 
among  the  angles  inversely  as  the  "  weights."  1  On  the  United 
States  Coast  Survey,  in  six  triangles  measured  in  1844  by 
Prof.  Bache,  the  greatest  error  was  six-tenths  of  a  second. 

(246.)  Interior  Filling-up.  The  stations  whose  positions 
have  been  determined  by  the  triangulation  are  so  many  fixed 
points,  from  which  more  minute  surveys  may  start  and  inter- 
polate any  other  points.  The  trigonometrical  points  are  like 
the  observed  latitudes  and  longitudes  which  the  mariner  ob- 
tains at  every  opportunity,  so  as  to  take  a  new  departure  from 
them,  and  determine  his  course  in  the  intervals  by  the  less  pre- 
cise methods  of  his  compass  and  log.  The  chief  interior  points 
may  be  obtained  by  "  secondary  triangulation,"  and  the  minor 
details  be  then  filled  in  by  any  of  the  methods  of  surveying, 
with  chain,  compass,  or  transit,  already  explained,  or  by  the 
plane-table. 

With  the  transit  or  theodolite,  "traversing"  is  the  best 
mode  of  surveying,  the  instrument  being  set  at  zero,  and  being 

1  L.  S.,  Art.  (369). 


CALCULATING  THE  SIDES  OF  THE  TRIANGLES. 


165 


then  directed  from  one  of  the  trigonometrical  points  to 
another,  which  line  therefore  becomes  the  "  Meridian  "  of  that 
survey.  On  reaching  this  second  point,  in  the  course  of  the 
survey,  and  sighting  back  to  the  first,  the  reading  should  of 
course  be  0°. 


CHAPTER    II. 


CALCULATING     THE     SIDES     OF     THE     TRIANGLES. 

(247.)  ONE  side  of  a  spherical  triangle  having  been  meas- 
ured or  calculated,  and  all  the  angles  observed,  the  other  sides 
can  be  computed  by  employing  the  principles  of  spherical 
trigonometry.  This,  however,  is  very  laborious,  and  other 
methods  have  been  adopted  which,  with  less  work,  give  results 
equally  accurate. 

(248.)     Delambre's    Method.  FlG  m 

Imagine  the  three  angular 
points  of  each  spherical  tri- 
angle to  be  joined  by  straight 
lines,  chords  of  the  arc,  so  as  to 
form  a  plane  triangle,  as  in 
Fig.  209.  Reduce  the  given 
curved  side  to  its  chord,  and 
the  spherical  angles  to  the 
plane  angles  contained  by  these 
chords.  Compute  the  other 
sides  or  chords  by  plane  trigonometry,  and  then  calculate  the 
arcs  corresponding  to  them. 

I     To  reduce  any  arc  to  its  chord  we  have : 

Chord  of  an  arc  a  =  2  sin.  -J  a. 
Or,  if  a  be  the  arc  in  terms  of  the  radius : 
Chord  of  a  =  a  —  -     a*. 


166      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

To  reduce  an  angle  of  a  spherical  triangle  to  the  corre- 
sponding angle  between  the  chords  of  the  including  arcs : 

Let  ABC,  Fig.  210,  be  a  spherical  triangle,  and  O  the 
centre  of  the  sphere.  It  is  required  to  reduce  the  spherical 
angle  at  A  to  the  plane  angle  between 
the  chords  A  B  and  A  C.  Draw  O  G 
and  O  H  parallel  to  A  B  and  A  C,  and 
prolong  the  arcs  to  G  and  H.  The 
lines  O  D  and  O  E,  bisecting  the  chords 
A  B  and  A  C,  will  be  perpendicular  to 
them,  and  also  to  O  G  and  O  H.  Then 
D  G  and  E  H  are  quadrants.  ISTow,  in 
the  spherical  triangle  A  G  H,  having 
the  arcs  A  G  and  A  H,  and  the  included 
angle  G  A  H,  the  measure  of  the  angle  G  O  H  is  found  by 
spherical  trigonometry.  But  G  O  H  =  B  A  C,  the  required 
angle. 

The  sum  of  the  three  plane  angles  thus  found  will  be 
equal  to  two  right  angles,  if  the  observations  of  the  spherical 
angles  and  the  work  of  reducing  have  been  correctly  done. 

(249.)  Legendre's  Method.  His  theorem  is  this:  "In  any 
spherical  triangle,  the  sides  of  which  are  very  small  compared 
to  the  radius  of  the  sphere,  if  each  of  the  angles  be  diminished 
by  -J  of  the  true  spherical  excess,  the  sines  of  these  angles  will 
be  proportioned  to  the  lengths  of  the  opposite  sides ;  and  the 
triangle  may  therefore  be  calculated  as  if  it  were  a  plane  one." 

This  is  the  easiest  method. 

All  three  methods  were  used  for  the  French  "Base  du 
systeme  mefotyue" 

In  the  British  "  Ordnance  Survey "  the  triangles  were 
mostly  calculated  by  the  second  method,  and  checked  by  the 
third. 

The  difference  on  100  miles  is  only  a  fraction  of  a  yard. 

(250.)  Co-ordinates  of  the  Points.  The  polar  spherical  co- 
ordinates of  a  point  with  respect  to  another  point  are  these : 

Ml;       fLQjt'*~*-*-*-*3 


r 


» 


\    -v 


CALCULATING  THE  SIDES  OF  THE  TRIANGLES.  167 

the  length  of  the  arc  of  the  great  circle  passing  through  the 
points,  and  its  azimuth,  i.e.,  the  angle  it  makes  with  the 
meridian  passing  through  one  of  its  points. 

The  rectangular  spherical  coordinates  of  a  point  have  for 
axes  the  meridian  passing  through  the  origin,  and  a  per- 
pendicular to  it.  For  short  distances  these  may  be  regarded 
as  in  one  plane.  For  greater  distances  new  meridians  must 
be  taken — say,  not  farther  apart  than  50  miles. 

"Within  that  limit  the  successive  triangles  may  be  con- 
ceived to  be  turned  down  into  the  same  plane. 

The  astronomical  coordinates  of  a  point  are  its  latitude 
and  longitude.  These  are  determined  by  practical  astronomy. 

The  transformation  of  these  coordinates  to  polar  or  rec- 
tangular, and  vice  versa,  is  very  important.  It  is  done  by 
spherical  trigonometry.  The  latitude  and  longitude  of  any 
one  point  are  very  accurately  determined  by  the  mean  of  a 
great  number  of  astronomical  observations,  and  those  of  the 
other  points  are  calculated  from  these.  Those  of  some  other 
points  may  be  observed  as  checks. 

It  is  found  that  the  observed  and  calculated  latitude  and 
longitude  of  a  place  do  not  always  agree,  even  when  the  earth 
is  considered  as  an  ellipsoid  of  revo- 
lution ;  in  consequence  of  the  irreg- 
ularity of  the  form  of  the  earth.  The 
difference  of  the  "geodesic"  from 
the  astronomical  determination  of 
difference  of  latitude  and  longitude,  ^ 
is  called  the  "  station  error."  £/ 

A  "  geodesic  line  "  is  the  short- 
est line  which  can  be  drawn  on  the 
ellipsoid,  corresponding  to  an  arc 
of  a  great  circle  on  the  sphere.  It 
is  the  line  of  least  curvature. 

(251.)  PEOB.  1.   Given  latitude  TE  =  latitude, 

and  longitude  of  A,  and  the  azimuth  and  distance  from  A  to  B. 
Required  the  latitude  and  longitude  of  B,  and  the  azimuth 


168      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

from  B  to  A.  The  distance  is  measured  on  the  arc  of  a  great 
circle  passing  through  those  points,  the  earth  being  assumed 
to  be  a  sphere. 

We  have  given  two  sides  and  the  included  angle,  to  find 
the  remaining  parts. 

By  spherical  trigonometry  we  have  : 


The  azimuth  from  B  to  A  =  B  =  J(B  +  P)  +  i(B  -  P). 
The   difference  of  long.   =  P  =  J(B  +  P)  -  i(B  -  P). 
To  find  the  co-latitude  of  B  =  P  B,  we  have  : 

tan.  4  P  B  =  tan.HAP-AB)Sin.HB  +  P) 
sin.  t(-B  —  P) 


(252.)    Otherwise.—  Let  fall  from  B, 
B  C  perpendicular  to  A  P.     Then, 

tang.  A  C  =  tang.  A  B  .  cos.  P  A  B. 
PC=PA-AC. 


. 
"     D      cos.  A  B .  cos.  PC    b  f  i'  * 

• I        cos.  AC. — •  r 


COS 

sin.  A  .  sin.  A  B 
"Tin.  PB~ 

sin.  A  .  sin.  A  P 


sin.  A  P  B  = 

sin.  ABP  = 
(253.)  PEOB.  2.  Given  latitude  and  longitude  of  A  and  B 


CALCULATING  THE  SIDES  OF  THE  TRIANGLES. 


169 


to  find  the  distance  between 
them  and  the  azimuth  from 
each  to  the  other ;  i.  e.,  to  find 
the  length  and  direction  of  the 
arc  of  a  great  circle  passing 
through  those  points. 

The  angle  of  P  is  the  differ- 
ence of  longitude,  P  B  is  the 
co-latitude  of  B,  and  P  A  is  the 
co-latitude  of  A.  Then  we  have 
two  sides  and  the  included 
angle  to  find  the  remaining 
parts. 


Fio,  213. 


tan.i(PA-PB)Bin.i(B  +  A) 

I  till,  -a-  Lex  J_>  —  -  ;  -  -  —  r^pr  --  T~T  —  -  . 

sin.  £  (  B  —  A) 

This  is  strictly  a  case  of  spherical  location,  required  in 
planning  a  road  between  two  distant  points,  and  in  navigating 
a  vessel. 

The  distance  may  also  be  found  thus  : 

Let  a  and  j3  represent  the  co-latitudes  of  A  and  B. 

cos.  A  B  =  cos.  a  .  cos.  /3  -f  sin.  a  .  sin.  (3  .  cos.  P. 
Put  tang.  0  =  tan.  a  .  cos.  P  ; 

„-  cos.  a  .  cos.(j3  —  <f>) 

Then,  cos.  A  B  =  -         -  U 


^  •*+,+. 


170       LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

Any  other  sets  of  three  parts  of  the  triangle  P  A  B  being 
given,  the  rest  can  be  found  by  spherical  trigonometry. 

(254.)  For  great  accuracy,  the  earth  must  be  regarded  as  a 
spheroid.  The  following  formulas  for  computing  the  geodesic 
latitudes,  longitudes,  and  azimuths  of  points  of  a  triangula- 
tion,  are  from  Captain  Lee's  Tables  and  Formulas  : 

Let  K  =  distance  in  yards  between  two  stations,  the  lat- 
itude and  longitude  of  one  of  which  are  known,  and  u"  this 
same  distance  converted  to  second  of  arc. 

L  =  latitude  of  first  station. 

M  =  longitude  of  first,  -f  if  west. 

Z  ==  azimuth  of  second  station  at  first,  counted  from  the 
south  around,  by  the  west,  from  0°  to  360°.  The  algebraic 
signs  of  the  sine  and  cosine  of  this  angle  must  be  carefully 
attended  to. 

L',  M',  Z',  the  same  things  at  second  station,  or  quantities 
required. 

a  =  the  equatorial  radius. 

e  —  the  eccentricity  =  0.0817  =  y  (- — s — )• 

K  =  the  radius  of  curvature  of  the  meridian,  in  yards. 
N  =  the  radius  of  curvature  of  a  section  perpendicular  to 
the  meridian,  in  yards. 

K          .  Kl- 


in.  V  a  sin.  1* 


sn 


L'  -  L-  (  1  +  e9  cos.3  l)u*  cos.  Z  -  (1  +  e*  cos.3  L) 
(i/sin.  Z)3  tan.  L  x  $  sin.  V. 

u"  sin.  Z 


cos. 


.  Z'  =  180°+  Z  -         *     Bin.  i(L  +  I/),  or 

Z'=180°+Z  —  (w'sin.  Ztan.L+  u"  sin.  Zoos.  Z£sin.  1*). 


CALCULATING  THE  SIDES  OF  THE  TRIANGLE. 


171 


The  quantity 


-     ^ 
cos.  -i 


sin.  £  (  L  +  I/),  or  (  M'—  M  )  sin. 


(L  +  I/),  by  which  the  azimuth  at  one  end  of  a  line  exceeds 
the  azimuth  at  the  other,  is  called  the  convergence  of  the 
meridians. 

In  terms  of  the  coordinates  of  rectangular  axes  referred  to 
one  of  the  points  of  the  triangulation,  the  latitude  and  longi- 
tude of  which  are  known,  y  being  the  ordinate  in  the  direction 
of  the  meridian,  and  x  the  ordinate  perpendicular  to  it  : 


-L± 


•  tan- 


7=  270°  ±  ^ 


a? 


sin.  I' 


cos.  I/" 


tan.  L7. 


(255.)  Calculation  of  a  spherical  triangle  by  Legendre's 
method.  (Art.  249.) 

The  following  example  is  from  the  United  States  Coast 
Survey : 


No. 
1 

Denomi- 
nation. 

Observed 
angles. 

Correc- 
tion. 

Spherical 
angles. 

Spherical 
excess. 

Plane,  angles, 
and  distances. 

Loga- 
rithms. 

Prince.  . 
Buck... 
Hill.... 

0      '        » 
41  47  41.79 
81  13  13.78 
56  59  07.39 

Buckt 
Prince 
Prince 

-o!60 
—  0.60 
—0.60 

o  Hill 

4L19 
13.18 
06.79 

0^39 
0.39 
0.38 

°      i         „ 
41  47  40.80 
81  13  12.79 
56  59  06.41 
m. 
19189.80 
28456.10 
24144.18 

0.1762239 
9.9948811 
9.9235180 

4.2830705 
4.4541755 
4.3828124 

to  Hill  

to  Buck.  . 

The  data  for  calculation  are  one  side  (Buck  to  Hill),  and 
the  observed  angles. 

To  determine  the  spherical  excess,  apply  the  formula  given 
in  Art.  244  : 

648000" 


As  the  surface,  s,  of  the  triangle  is  very  small,  compared 


172      LEVELLING,  TOPOGRAPHY,  AND  HIGHER  SURVEYING. 

with  the  diameter  of  the  earth,  it  may  be,obtained  with  suf- 
ficient accuracy,  for  this  purpose,  by  treating  the  triangle  as 
if  it  were  plane.  Then,  the  three  angles  and  one  side  being 
given,  we  have  the  formula : 

,   3Sin.  B.     Sin.  C 

s  =  i a  -    -^-. r > 

Sin.  A. 

in  which  a  is  the  given  side ;  B  and  C,  the  adjacent  angles, 
and  A,  the  opposite  angle. 

In  getting  the  value  of  the  fraction  in  the  formula  for  the 
spherical  excess,  the  radius  of  the  earth,  r,  must  be  taken  in 
the  same  unit  of  measure  as  s.  The  values  used  on  the  Coast 
Survey  are :  Equatorial  radius,  6377397.16  metres ;  polar  ra- 
dius, 6356078.96  metres ;  and  mean  radius,  6366738.06  metres. 

To  find  Log.  s. 

Log.  £  =  1.6989700 

Log.  a*  (19189.80)'  =  8.5661410 

o         i  a 

Log.  sin.  B  (81  13  13.78)  =  9.9948813 
Log.  sin.  C  (56  59  07.39)  =  9.9235193 
Co-log,  sin.  A  (41  47  41.79)  =  0.1762216 

Log.  s.  =  8.3597332 

648000" 
To  find  Log. — 

Log.  648000"  =  5.8115750 
Co-log.  TT  (3.1415927)  =  9.5028501 
Co-log,  r*  (6366738.06)2  =  6.3921660 


Log.  =9.7065911 

7T  T 

Log.  s.  =  8.3597332 

Log.  e,  spherical  excess  =    .0663243 

Spherical  excess        =  1//.16 

The  difference  between  the  sum  of  the  observed  angles 
and  180°  plus  the  spherical  excess  (1M6),  is  1".80,  which 
will  make  a  correction  for  each  angle  of  0".60.  Placing  this 


CALCULATING  THE  SIDES  OF  THE  TRIANGLE. 


173 


correction  in  the  fourth  column,  and  subtracting  it  from  the 
observed  angles,  we  get  the  corrected  spherical  angles  for  the 
fifth  column.  One-third  of  the  spherical  excess  (sixth  column) 
is  then  subtracted  from  the  spherical  angles  to  reduce  them  to 
plane  angles,  which  are  placed  in  the  seventh  column.  Using 
these  plane  angles,  and  the  given  side,  and  applying  the  sine 
proportion,  we  have : 


To  find  b. 

Log.  a  =  4.2830705 
Log.  sin.  B  =  9.9948811 
Co-log,  sin.  A  =  0.1762239 

Log.  I     •=  4.4541755 
Prince  to  Hill  =  28456.10 


To  find  c. 

Log.  a  =  4.2830705 

Log.  sin.  C  =  9.9235180 

Co-log,  sin.  A      =  0.1762239 

Log.  c  =  4.3828124 

Prince  to  Buck  =  24144.18 


The  logarithms  of  the  sides  and  of  the  sines  of  the  plane 
angles  are  placed  in  the  last  column.  For  convenience  in  cal- 
culation, the  co-log,  of  angle  opposite  the  given  side  is  taken. 


THE     END. 


-**<- 


^.< 


^t.:e;.V- 


ijU  _L 


'?*«t*f 


/  ^-'  /•'    / 
£,  ot.  cL*. 


/ 


t   M 


'' 


UNIVERSITY  OP  CALIFORNIA  LIBRARY 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


'     .. 


/ 


61954LU 


30W-6/14 


.//'/          :/-, 


v£C^  c?*-£4/  iu^fi 


